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Properties

Label 275.4.a.a.1.1
Level $275$
Weight $4$
Character 275.1
Self dual yes
Analytic conductor $16.226$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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] = [275,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,3] 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: \(16.2255252516\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} -3.00000 q^{6} +9.00000 q^{7} +15.0000 q^{8} -18.0000 q^{9} +11.0000 q^{11} -21.0000 q^{12} -2.00000 q^{13} -9.00000 q^{14} +41.0000 q^{16} -21.0000 q^{17} +18.0000 q^{18} -85.0000 q^{19} +27.0000 q^{21} -11.0000 q^{22} -22.0000 q^{23} +45.0000 q^{24} +2.00000 q^{26} -135.000 q^{27} -63.0000 q^{28} -165.000 q^{29} -83.0000 q^{31} -161.000 q^{32} +33.0000 q^{33} +21.0000 q^{34} +126.000 q^{36} -1.00000 q^{37} +85.0000 q^{38} -6.00000 q^{39} -478.000 q^{41} -27.0000 q^{42} +8.00000 q^{43} -77.0000 q^{44} +22.0000 q^{46} -126.000 q^{47} +123.000 q^{48} -262.000 q^{49} -63.0000 q^{51} +14.0000 q^{52} +683.000 q^{53} +135.000 q^{54} +135.000 q^{56} -255.000 q^{57} +165.000 q^{58} -290.000 q^{59} +257.000 q^{61} +83.0000 q^{62} -162.000 q^{63} -167.000 q^{64} -33.0000 q^{66} -776.000 q^{67} +147.000 q^{68} -66.0000 q^{69} -313.000 q^{71} -270.000 q^{72} -902.000 q^{73} +1.00000 q^{74} +595.000 q^{76} +99.0000 q^{77} +6.00000 q^{78} +830.000 q^{79} +81.0000 q^{81} +478.000 q^{82} -842.000 q^{83} -189.000 q^{84} -8.00000 q^{86} -495.000 q^{87} +165.000 q^{88} +25.0000 q^{89} -18.0000 q^{91} +154.000 q^{92} -249.000 q^{93} +126.000 q^{94} -483.000 q^{96} +1784.00 q^{97} +262.000 q^{98} -198.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.353553 −0.176777 − 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) 3.00000 0.577350 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) −7.00000 −0.875000
\(5\) 0 0
\(6\) −3.00000 −0.204124
\(7\) 9.00000 0.485954 0.242977 − 0.970032i \(-0.421876\pi\)
0.242977 + 0.970032i \(0.421876\pi\)
\(8\) 15.0000 0.662913
\(9\) −18.0000 −0.666667
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) −21.0000 −0.505181
\(13\) −2.00000 −0.0426692 −0.0213346 − 0.999772i \(-0.506792\pi\)
−0.0213346 + 0.999772i \(0.506792\pi\)
\(14\) −9.00000 −0.171811
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) −21.0000 −0.299603 −0.149801 − 0.988716i \(-0.547863\pi\)
−0.149801 + 0.988716i \(0.547863\pi\)
\(18\) 18.0000 0.235702
\(19\) −85.0000 −1.02633 −0.513167 − 0.858289i \(-0.671528\pi\)
−0.513167 + 0.858289i \(0.671528\pi\)
\(20\) 0 0
\(21\) 27.0000 0.280566
\(22\) −11.0000 −0.106600
\(23\) −22.0000 −0.199449 −0.0997243 − 0.995015i \(-0.531796\pi\)
−0.0997243 + 0.995015i \(0.531796\pi\)
\(24\) 45.0000 0.382733
\(25\) 0 0
\(26\) 2.00000 0.0150859
\(27\) −135.000 −0.962250
\(28\) −63.0000 −0.425210
\(29\) −165.000 −1.05654 −0.528271 − 0.849076i \(-0.677160\pi\)
−0.528271 + 0.849076i \(0.677160\pi\)
\(30\) 0 0
\(31\) −83.0000 −0.480879 −0.240439 − 0.970664i \(-0.577292\pi\)
−0.240439 + 0.970664i \(0.577292\pi\)
\(32\) −161.000 −0.889408
\(33\) 33.0000 0.174078
\(34\) 21.0000 0.105926
\(35\) 0 0
\(36\) 126.000 0.583333
\(37\) −1.00000 −0.00444322 −0.00222161 − 0.999998i \(-0.500707\pi\)
−0.00222161 + 0.999998i \(0.500707\pi\)
\(38\) 85.0000 0.362864
\(39\) −6.00000 −0.0246351
\(40\) 0 0
\(41\) −478.000 −1.82076 −0.910379 − 0.413776i \(-0.864210\pi\)
−0.910379 + 0.413776i \(0.864210\pi\)
\(42\) −27.0000 −0.0991950
\(43\) 8.00000 0.0283718 0.0141859 − 0.999899i \(-0.495484\pi\)
0.0141859 + 0.999899i \(0.495484\pi\)
\(44\) −77.0000 −0.263822
\(45\) 0 0
\(46\) 22.0000 0.0705157
\(47\) −126.000 −0.391042 −0.195521 − 0.980699i \(-0.562640\pi\)
−0.195521 + 0.980699i \(0.562640\pi\)
\(48\) 123.000 0.369865
\(49\) −262.000 −0.763848
\(50\) 0 0
\(51\) −63.0000 −0.172976
\(52\) 14.0000 0.0373356
\(53\) 683.000 1.77014 0.885069 − 0.465461i \(-0.154111\pi\)
0.885069 + 0.465461i \(0.154111\pi\)
\(54\) 135.000 0.340207
\(55\) 0 0
\(56\) 135.000 0.322145
\(57\) −255.000 −0.592554
\(58\) 165.000 0.373544
\(59\) −290.000 −0.639912 −0.319956 − 0.947432i \(-0.603668\pi\)
−0.319956 + 0.947432i \(0.603668\pi\)
\(60\) 0 0
\(61\) 257.000 0.539434 0.269717 − 0.962940i \(-0.413070\pi\)
0.269717 + 0.962940i \(0.413070\pi\)
\(62\) 83.0000 0.170016
\(63\) −162.000 −0.323970
\(64\) −167.000 −0.326172
\(65\) 0 0
\(66\) −33.0000 −0.0615457
\(67\) −776.000 −1.41498 −0.707489 − 0.706725i \(-0.750172\pi\)
−0.707489 + 0.706725i \(0.750172\pi\)
\(68\) 147.000 0.262152
\(69\) −66.0000 −0.115152
\(70\) 0 0
\(71\) −313.000 −0.523187 −0.261593 − 0.965178i \(-0.584248\pi\)
−0.261593 + 0.965178i \(0.584248\pi\)
\(72\) −270.000 −0.441942
\(73\) −902.000 −1.44618 −0.723090 − 0.690754i \(-0.757279\pi\)
−0.723090 + 0.690754i \(0.757279\pi\)
\(74\) 1.00000 0.00157091
\(75\) 0 0
\(76\) 595.000 0.898042
\(77\) 99.0000 0.146521
\(78\) 6.00000 0.00870982
\(79\) 830.000 1.18205 0.591027 − 0.806652i \(-0.298723\pi\)
0.591027 + 0.806652i \(0.298723\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 478.000 0.643735
\(83\) −842.000 −1.11351 −0.556756 − 0.830676i \(-0.687954\pi\)
−0.556756 + 0.830676i \(0.687954\pi\)
\(84\) −189.000 −0.245495
\(85\) 0 0
\(86\) −8.00000 −0.0100310
\(87\) −495.000 −0.609995
\(88\) 165.000 0.199876
\(89\) 25.0000 0.0297752 0.0148876 − 0.999889i \(-0.495261\pi\)
0.0148876 + 0.999889i \(0.495261\pi\)
\(90\) 0 0
\(91\) −18.0000 −0.0207353
\(92\) 154.000 0.174517
\(93\) −249.000 −0.277635
\(94\) 126.000 0.138254
\(95\) 0 0
\(96\) −483.000 −0.513500
\(97\) 1784.00 1.86740 0.933700 − 0.358057i \(-0.116561\pi\)
0.933700 + 0.358057i \(0.116561\pi\)
\(98\) 262.000 0.270061
\(99\) −198.000 −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 275.4.a.a.1.1 1
3.2 odd 2 2475.4.a.h.1.1 1
5.2 odd 4 275.4.b.a.199.1 2
5.3 odd 4 275.4.b.a.199.2 2
5.4 even 2 55.4.a.a.1.1 ✓ 1
15.14 odd 2 495.4.a.a.1.1 1
20.19 odd 2 880.4.a.j.1.1 1
55.54 odd 2 605.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.4.a.a.1.1 ✓ 1 5.4 even 2
275.4.a.a.1.1 1 1.1 even 1 trivial
275.4.b.a.199.1 2 5.2 odd 4
275.4.b.a.199.2 2 5.3 odd 4
495.4.a.a.1.1 1 15.14 odd 2
605.4.a.b.1.1 1 55.54 odd 2
880.4.a.j.1.1 1 20.19 odd 2
2475.4.a.h.1.1 1 3.2 odd 2