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Properties

Label 2790.2.a.be.1.2
Level $2790$
Weight $2$
Character 2790.1
Self dual yes
Analytic conductor $22.278$
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] = [2790,2,Mod(1,2790)] 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("2790.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2790, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2790.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,2,-2,0,1,-2,0,2,-1,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(22.2782621639\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 2790.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.56155 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.56155 q^{11} +2.00000 q^{13} -2.56155 q^{14} +1.00000 q^{16} +3.12311 q^{17} -7.68466 q^{19} -1.00000 q^{20} +2.56155 q^{22} -1.43845 q^{23} +1.00000 q^{25} -2.00000 q^{26} +2.56155 q^{28} -7.12311 q^{29} +1.00000 q^{31} -1.00000 q^{32} -3.12311 q^{34} -2.56155 q^{35} -3.12311 q^{37} +7.68466 q^{38} +1.00000 q^{40} -7.12311 q^{41} +12.8078 q^{43} -2.56155 q^{44} +1.43845 q^{46} -5.12311 q^{47} -0.438447 q^{49} -1.00000 q^{50} +2.00000 q^{52} -7.43845 q^{53} +2.56155 q^{55} -2.56155 q^{56} +7.12311 q^{58} +13.1231 q^{59} +6.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -15.3693 q^{67} +3.12311 q^{68} +2.56155 q^{70} +7.68466 q^{71} -10.8078 q^{73} +3.12311 q^{74} -7.68466 q^{76} -6.56155 q^{77} -4.31534 q^{79} -1.00000 q^{80} +7.12311 q^{82} -14.2462 q^{83} -3.12311 q^{85} -12.8078 q^{86} +2.56155 q^{88} +13.6847 q^{89} +5.12311 q^{91} -1.43845 q^{92} +5.12311 q^{94} +7.68466 q^{95} -6.00000 q^{97} +0.438447 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} + 4 q^{13} - q^{14} + 2 q^{16} - 2 q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - 7 q^{23} + 2 q^{25} - 4 q^{26} + q^{28} - 6 q^{29}+ \cdots + 5 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.56155 0.968176 0.484088 − 0.875019i \(-0.339151\pi\)
0.484088 + 0.875019i \(0.339151\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −2.56155 −0.772337 −0.386169 − 0.922428i \(-0.626202\pi\)
−0.386169 + 0.922428i \(0.626202\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −2.56155 −0.684604
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.12311 0.757464 0.378732 − 0.925506i \(-0.376360\pi\)
0.378732 + 0.925506i \(0.376360\pi\)
\(18\) 0 0
\(19\) −7.68466 −1.76298 −0.881491 − 0.472201i \(-0.843460\pi\)
−0.881491 + 0.472201i \(0.843460\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 2.56155 0.546125
\(23\) −1.43845 −0.299937 −0.149968 − 0.988691i \(-0.547917\pi\)
−0.149968 + 0.988691i \(0.547917\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 2.56155 0.484088
\(29\) −7.12311 −1.32273 −0.661364 − 0.750065i \(-0.730022\pi\)
−0.661364 + 0.750065i \(0.730022\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.12311 −0.535608
\(35\) −2.56155 −0.432981
\(36\) 0 0
\(37\) −3.12311 −0.513435 −0.256718 − 0.966486i \(-0.582641\pi\)
−0.256718 + 0.966486i \(0.582641\pi\)
\(38\) 7.68466 1.24662
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −7.12311 −1.11244 −0.556221 − 0.831034i \(-0.687749\pi\)
−0.556221 + 0.831034i \(0.687749\pi\)
\(42\) 0 0
\(43\) 12.8078 1.95317 0.976583 − 0.215142i \(-0.0690213\pi\)
0.976583 + 0.215142i \(0.0690213\pi\)
\(44\) −2.56155 −0.386169
\(45\) 0 0
\(46\) 1.43845 0.212087
\(47\) −5.12311 −0.747282 −0.373641 − 0.927573i \(-0.621891\pi\)
−0.373641 + 0.927573i \(0.621891\pi\)
\(48\) 0 0
\(49\) −0.438447 −0.0626353
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −7.43845 −1.02175 −0.510875 − 0.859655i \(-0.670678\pi\)
−0.510875 + 0.859655i \(0.670678\pi\)
\(54\) 0 0
\(55\) 2.56155 0.345400
\(56\) −2.56155 −0.342302
\(57\) 0 0
\(58\) 7.12311 0.935310
\(59\) 13.1231 1.70848 0.854241 − 0.519877i \(-0.174022\pi\)
0.854241 + 0.519877i \(0.174022\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −1.00000 −0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −15.3693 −1.87766 −0.938830 − 0.344380i \(-0.888089\pi\)
−0.938830 + 0.344380i \(0.888089\pi\)
\(68\) 3.12311 0.378732
\(69\) 0 0
\(70\) 2.56155 0.306164
\(71\) 7.68466 0.912001 0.456001 − 0.889979i \(-0.349281\pi\)
0.456001 + 0.889979i \(0.349281\pi\)
\(72\) 0 0
\(73\) −10.8078 −1.26495 −0.632477 − 0.774580i \(-0.717962\pi\)
−0.632477 + 0.774580i \(0.717962\pi\)
\(74\) 3.12311 0.363054
\(75\) 0 0
\(76\) −7.68466 −0.881491
\(77\) −6.56155 −0.747758
\(78\) 0 0
\(79\) −4.31534 −0.485514 −0.242757 − 0.970087i \(-0.578052\pi\)
−0.242757 + 0.970087i \(0.578052\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 7.12311 0.786615
\(83\) −14.2462 −1.56372 −0.781862 − 0.623451i \(-0.785730\pi\)
−0.781862 + 0.623451i \(0.785730\pi\)
\(84\) 0 0
\(85\) −3.12311 −0.338748
\(86\) −12.8078 −1.38110
\(87\) 0 0
\(88\) 2.56155 0.273062
\(89\) 13.6847 1.45057 0.725285 − 0.688448i \(-0.241708\pi\)
0.725285 + 0.688448i \(0.241708\pi\)
\(90\) 0 0
\(91\) 5.12311 0.537047
\(92\) −1.43845 −0.149968
\(93\) 0 0
\(94\) 5.12311 0.528408
\(95\) 7.68466 0.788429
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0.438447 0.0442899
\(99\) 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 2790.2.a.be.1.2 2
3.2 odd 2 930.2.a.p.1.2 ✓ 2
12.11 even 2 7440.2.a.bl.1.1 2
15.2 even 4 4650.2.d.bd.3349.4 4
15.8 even 4 4650.2.d.bd.3349.1 4
15.14 odd 2 4650.2.a.ce.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.p.1.2 ✓ 2 3.2 odd 2
2790.2.a.be.1.2 2 1.1 even 1 trivial
4650.2.a.ce.1.1 2 15.14 odd 2
4650.2.d.bd.3349.1 4 15.8 even 4
4650.2.d.bd.3349.4 4 15.2 even 4
7440.2.a.bl.1.1 2 12.11 even 2