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Properties

Label 2790.2.a.be.1.1
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 1.1
Root \(-1.56155\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.56155 q^{11} +2.00000 q^{13} +1.56155 q^{14} +1.00000 q^{16} -5.12311 q^{17} +4.68466 q^{19} -1.00000 q^{20} -1.56155 q^{22} -5.56155 q^{23} +1.00000 q^{25} -2.00000 q^{26} -1.56155 q^{28} +1.12311 q^{29} +1.00000 q^{31} -1.00000 q^{32} +5.12311 q^{34} +1.56155 q^{35} +5.12311 q^{37} -4.68466 q^{38} +1.00000 q^{40} +1.12311 q^{41} -7.80776 q^{43} +1.56155 q^{44} +5.56155 q^{46} +3.12311 q^{47} -4.56155 q^{49} -1.00000 q^{50} +2.00000 q^{52} -11.5616 q^{53} -1.56155 q^{55} +1.56155 q^{56} -1.12311 q^{58} +4.87689 q^{59} +6.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} +9.36932 q^{67} -5.12311 q^{68} -1.56155 q^{70} -4.68466 q^{71} +9.80776 q^{73} -5.12311 q^{74} +4.68466 q^{76} -2.43845 q^{77} -16.6847 q^{79} -1.00000 q^{80} -1.12311 q^{82} +2.24621 q^{83} +5.12311 q^{85} +7.80776 q^{86} -1.56155 q^{88} +1.31534 q^{89} -3.12311 q^{91} -5.56155 q^{92} -3.12311 q^{94} -4.68466 q^{95} -6.00000 q^{97} +4.56155 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} + 4 q^{13} - q^{14} + 2 q^{16} - 2 q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - 7 q^{23} + 2 q^{25} - 4 q^{26} + q^{28} - 6 q^{29}+ \cdots + 5 q^{98}+O(q^{100}) \) 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
\(7\) −1.56155 −0.590211 −0.295106 − 0.955465i \(-0.595355\pi\)
−0.295106 + 0.955465i \(0.595355\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 1.56155 0.470826 0.235413 − 0.971895i \(-0.424356\pi\)
0.235413 + 0.971895i \(0.424356\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.56155 0.417343
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.12311 −1.24254 −0.621268 − 0.783598i \(-0.713382\pi\)
−0.621268 + 0.783598i \(0.713382\pi\)
\(18\) 0 0
\(19\) 4.68466 1.07473 0.537367 − 0.843348i \(-0.319419\pi\)
0.537367 + 0.843348i \(0.319419\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.56155 −0.332924
\(23\) −5.56155 −1.15966 −0.579832 − 0.814736i \(-0.696882\pi\)
−0.579832 + 0.814736i \(0.696882\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) −1.56155 −0.295106
\(29\) 1.12311 0.208555 0.104278 − 0.994548i \(-0.466747\pi\)
0.104278 + 0.994548i \(0.466747\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 5.12311 0.878605
\(35\) 1.56155 0.263951
\(36\) 0 0
\(37\) 5.12311 0.842233 0.421117 − 0.907006i \(-0.361638\pi\)
0.421117 + 0.907006i \(0.361638\pi\)
\(38\) −4.68466 −0.759952
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 1.12311 0.175400 0.0876998 − 0.996147i \(-0.472048\pi\)
0.0876998 + 0.996147i \(0.472048\pi\)
\(42\) 0 0
\(43\) −7.80776 −1.19067 −0.595336 − 0.803477i \(-0.702981\pi\)
−0.595336 + 0.803477i \(0.702981\pi\)
\(44\) 1.56155 0.235413
\(45\) 0 0
\(46\) 5.56155 0.820006
\(47\) 3.12311 0.455552 0.227776 − 0.973714i \(-0.426855\pi\)
0.227776 + 0.973714i \(0.426855\pi\)
\(48\) 0 0
\(49\) −4.56155 −0.651650
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −11.5616 −1.58810 −0.794051 − 0.607852i \(-0.792032\pi\)
−0.794051 + 0.607852i \(0.792032\pi\)
\(54\) 0 0
\(55\) −1.56155 −0.210560
\(56\) 1.56155 0.208671
\(57\) 0 0
\(58\) −1.12311 −0.147471
\(59\) 4.87689 0.634918 0.317459 − 0.948272i \(-0.397170\pi\)
0.317459 + 0.948272i \(0.397170\pi\)
\(60\) 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
\(67\) 9.36932 1.14464 0.572322 − 0.820029i \(-0.306043\pi\)
0.572322 + 0.820029i \(0.306043\pi\)
\(68\) −5.12311 −0.621268
\(69\) 0 0
\(70\) −1.56155 −0.186641
\(71\) −4.68466 −0.555967 −0.277983 − 0.960586i \(-0.589666\pi\)
−0.277983 + 0.960586i \(0.589666\pi\)
\(72\) 0 0
\(73\) 9.80776 1.14791 0.573956 − 0.818886i \(-0.305408\pi\)
0.573956 + 0.818886i \(0.305408\pi\)
\(74\) −5.12311 −0.595549
\(75\) 0 0
\(76\) 4.68466 0.537367
\(77\) −2.43845 −0.277887
\(78\) 0 0
\(79\) −16.6847 −1.87717 −0.938585 − 0.345047i \(-0.887863\pi\)
−0.938585 + 0.345047i \(0.887863\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −1.12311 −0.124026
\(83\) 2.24621 0.246554 0.123277 − 0.992372i \(-0.460660\pi\)
0.123277 + 0.992372i \(0.460660\pi\)
\(84\) 0 0
\(85\) 5.12311 0.555679
\(86\) 7.80776 0.841933
\(87\) 0 0
\(88\) −1.56155 −0.166462
\(89\) 1.31534 0.139426 0.0697130 − 0.997567i \(-0.477792\pi\)
0.0697130 + 0.997567i \(0.477792\pi\)
\(90\) 0 0
\(91\) −3.12311 −0.327390
\(92\) −5.56155 −0.579832
\(93\) 0 0
\(94\) −3.12311 −0.322124
\(95\) −4.68466 −0.480636
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 4.56155 0.460786
\(99\) 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.1.1 2
3.2 odd 2 930.2.a.p.1.1 ✓ 2
12.11 even 2 7440.2.a.bl.1.2 2
15.2 even 4 4650.2.d.bd.3349.3 4
15.8 even 4 4650.2.d.bd.3349.2 4
15.14 odd 2 4650.2.a.ce.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.p.1.1 ✓ 2 3.2 odd 2
2790.2.a.be.1.1 2 1.1 even 1 trivial
4650.2.a.ce.1.2 2 15.14 odd 2
4650.2.d.bd.3349.2 4 15.8 even 4
4650.2.d.bd.3349.3 4 15.2 even 4
7440.2.a.bl.1.2 2 12.11 even 2