Properties

Label 2775.2.a.k.1.1
Level $2775$
Weight $2$
Character 2775.1
Self dual yes
Analytic conductor $22.158$
Analytic rank $1$
Dimension $2$
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] = [2775,2,Mod(1,2775)] 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("2775.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2775, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2775 = 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2775.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,-2,2,0,2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(22.1584865609\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2775.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} +2.41421 q^{6} -2.00000 q^{7} -4.41421 q^{8} +1.00000 q^{9} +2.82843 q^{11} -3.82843 q^{12} +4.82843 q^{13} +4.82843 q^{14} +3.00000 q^{16} -3.65685 q^{17} -2.41421 q^{18} -1.24264 q^{19} +2.00000 q^{21} -6.82843 q^{22} -4.41421 q^{23} +4.41421 q^{24} -11.6569 q^{26} -1.00000 q^{27} -7.65685 q^{28} -4.82843 q^{29} +6.00000 q^{31} +1.58579 q^{32} -2.82843 q^{33} +8.82843 q^{34} +3.82843 q^{36} -1.00000 q^{37} +3.00000 q^{38} -4.82843 q^{39} +0.656854 q^{41} -4.82843 q^{42} +1.24264 q^{43} +10.8284 q^{44} +10.6569 q^{46} -8.82843 q^{47} -3.00000 q^{48} -3.00000 q^{49} +3.65685 q^{51} +18.4853 q^{52} +2.65685 q^{53} +2.41421 q^{54} +8.82843 q^{56} +1.24264 q^{57} +11.6569 q^{58} +7.58579 q^{59} -12.0000 q^{61} -14.4853 q^{62} -2.00000 q^{63} -9.82843 q^{64} +6.82843 q^{66} +7.65685 q^{67} -14.0000 q^{68} +4.41421 q^{69} -7.31371 q^{71} -4.41421 q^{72} +11.4853 q^{73} +2.41421 q^{74} -4.75736 q^{76} -5.65685 q^{77} +11.6569 q^{78} +12.0711 q^{79} +1.00000 q^{81} -1.58579 q^{82} +1.65685 q^{83} +7.65685 q^{84} -3.00000 q^{86} +4.82843 q^{87} -12.4853 q^{88} -7.17157 q^{89} -9.65685 q^{91} -16.8995 q^{92} -6.00000 q^{93} +21.3137 q^{94} -1.58579 q^{96} -8.48528 q^{97} +7.24264 q^{98} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} - 2 q^{12} + 4 q^{13} + 4 q^{14} + 6 q^{16} + 4 q^{17} - 2 q^{18} + 6 q^{19} + 4 q^{21} - 8 q^{22} - 6 q^{23} + 6 q^{24} - 12 q^{26}+ \cdots + 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 − 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.82843 1.91421
\(5\) 0 0
\(6\) 2.41421 0.985599
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −4.41421 −1.56066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.82843 0.852803 0.426401 − 0.904534i \(-0.359781\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) −3.82843 −1.10517
\(13\) 4.82843 1.33916 0.669582 − 0.742738i \(-0.266473\pi\)
0.669582 + 0.742738i \(0.266473\pi\)
\(14\) 4.82843 1.29045
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −3.65685 −0.886917 −0.443459 − 0.896295i \(-0.646249\pi\)
−0.443459 + 0.896295i \(0.646249\pi\)
\(18\) −2.41421 −0.569036
\(19\) −1.24264 −0.285081 −0.142541 − 0.989789i \(-0.545527\pi\)
−0.142541 + 0.989789i \(0.545527\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) −6.82843 −1.45583
\(23\) −4.41421 −0.920427 −0.460214 − 0.887808i \(-0.652227\pi\)
−0.460214 + 0.887808i \(0.652227\pi\)
\(24\) 4.41421 0.901048
\(25\) 0 0
\(26\) −11.6569 −2.28610
\(27\) −1.00000 −0.192450
\(28\) −7.65685 −1.44701
\(29\) −4.82843 −0.896616 −0.448308 − 0.893879i \(-0.647973\pi\)
−0.448308 + 0.893879i \(0.647973\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.58579 0.280330
\(33\) −2.82843 −0.492366
\(34\) 8.82843 1.51406
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) −1.00000 −0.164399
\(38\) 3.00000 0.486664
\(39\) −4.82843 −0.773167
\(40\) 0 0
\(41\) 0.656854 0.102583 0.0512917 − 0.998684i \(-0.483666\pi\)
0.0512917 + 0.998684i \(0.483666\pi\)
\(42\) −4.82843 −0.745042
\(43\) 1.24264 0.189501 0.0947505 − 0.995501i \(-0.469795\pi\)
0.0947505 + 0.995501i \(0.469795\pi\)
\(44\) 10.8284 1.63245
\(45\) 0 0
\(46\) 10.6569 1.57127
\(47\) −8.82843 −1.28776 −0.643879 − 0.765127i \(-0.722676\pi\)
−0.643879 + 0.765127i \(0.722676\pi\)
\(48\) −3.00000 −0.433013
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 3.65685 0.512062
\(52\) 18.4853 2.56345
\(53\) 2.65685 0.364947 0.182473 − 0.983211i \(-0.441590\pi\)
0.182473 + 0.983211i \(0.441590\pi\)
\(54\) 2.41421 0.328533
\(55\) 0 0
\(56\) 8.82843 1.17975
\(57\) 1.24264 0.164592
\(58\) 11.6569 1.53062
\(59\) 7.58579 0.987585 0.493793 − 0.869580i \(-0.335610\pi\)
0.493793 + 0.869580i \(0.335610\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 − 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) −14.4853 −1.83963
\(63\) −2.00000 −0.251976
\(64\) −9.82843 −1.22855
\(65\) 0 0
\(66\) 6.82843 0.840521
\(67\) 7.65685 0.935434 0.467717 − 0.883878i \(-0.345077\pi\)
0.467717 + 0.883878i \(0.345077\pi\)
\(68\) −14.0000 −1.69775
\(69\) 4.41421 0.531409
\(70\) 0 0
\(71\) −7.31371 −0.867978 −0.433989 − 0.900918i \(-0.642894\pi\)
−0.433989 + 0.900918i \(0.642894\pi\)
\(72\) −4.41421 −0.520220
\(73\) 11.4853 1.34425 0.672125 − 0.740437i \(-0.265382\pi\)
0.672125 + 0.740437i \(0.265382\pi\)
\(74\) 2.41421 0.280647
\(75\) 0 0
\(76\) −4.75736 −0.545707
\(77\) −5.65685 −0.644658
\(78\) 11.6569 1.31988
\(79\) 12.0711 1.35810 0.679051 − 0.734091i \(-0.262392\pi\)
0.679051 + 0.734091i \(0.262392\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −1.58579 −0.175121
\(83\) 1.65685 0.181863 0.0909317 − 0.995857i \(-0.471016\pi\)
0.0909317 + 0.995857i \(0.471016\pi\)
\(84\) 7.65685 0.835431
\(85\) 0 0
\(86\) −3.00000 −0.323498
\(87\) 4.82843 0.517662
\(88\) −12.4853 −1.33094
\(89\) −7.17157 −0.760185 −0.380093 − 0.924948i \(-0.624108\pi\)
−0.380093 + 0.924948i \(0.624108\pi\)
\(90\) 0 0
\(91\) −9.65685 −1.01231
\(92\) −16.8995 −1.76189
\(93\) −6.00000 −0.622171
\(94\) 21.3137 2.19834
\(95\) 0 0
\(96\) −1.58579 −0.161849
\(97\) −8.48528 −0.861550 −0.430775 − 0.902459i \(-0.641760\pi\)
−0.430775 + 0.902459i \(0.641760\pi\)
\(98\) 7.24264 0.731617
\(99\) 2.82843 0.284268
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 2775.2.a.k.1.1 ✓ 2
3.2 odd 2 8325.2.a.bn.1.2 2
5.4 even 2 2775.2.a.p.1.2 yes 2
15.14 odd 2 8325.2.a.bg.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2775.2.a.k.1.1 ✓ 2 1.1 even 1 trivial
2775.2.a.p.1.2 yes 2 5.4 even 2
8325.2.a.bg.1.1 2 15.14 odd 2
8325.2.a.bn.1.2 2 3.2 odd 2