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Properties

Label 2790.2.d.d.559.2
Level $2790$
Weight $2$
Character 2790.559
Analytic conductor $22.278$
Analytic rank $1$
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] = [2790,2,Mod(559,2790)] 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("2790.559"); 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, 1, 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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,-4,0,0,0,0,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.2782621639\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 559.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2790.559
Dual form 2790.2.d.d.559.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +(-2.00000 - 1.00000i) q^{5} +2.00000i q^{7} -1.00000i q^{8} +(1.00000 - 2.00000i) q^{10} +2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +4.00000 q^{19} +(2.00000 + 1.00000i) q^{20} -2.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -2.00000 q^{26} -2.00000i q^{28} -10.0000 q^{29} -1.00000 q^{31} +1.00000i q^{32} +(2.00000 - 4.00000i) q^{35} +2.00000i q^{37} +4.00000i q^{38} +(-1.00000 + 2.00000i) q^{40} -2.00000 q^{41} +4.00000i q^{43} +2.00000 q^{46} +4.00000i q^{47} +3.00000 q^{49} +(-4.00000 + 3.00000i) q^{50} -2.00000i q^{52} -2.00000i q^{53} +2.00000 q^{56} -10.0000i q^{58} -6.00000 q^{59} -8.00000 q^{61} -1.00000i q^{62} -1.00000 q^{64} +(2.00000 - 4.00000i) q^{65} -8.00000i q^{67} +(4.00000 + 2.00000i) q^{70} +8.00000 q^{71} -10.0000i q^{73} -2.00000 q^{74} -4.00000 q^{76} -12.0000 q^{79} +(-2.00000 - 1.00000i) q^{80} -2.00000i q^{82} -4.00000i q^{83} -4.00000 q^{86} -14.0000 q^{89} -4.00000 q^{91} +2.00000i q^{92} -4.00000 q^{94} +(-8.00000 - 4.00000i) q^{95} +4.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 4 q^{5} + 2 q^{10} - 4 q^{14} + 2 q^{16} + 8 q^{19} + 4 q^{20} + 6 q^{25} - 4 q^{26} - 20 q^{29} - 2 q^{31} + 4 q^{35} - 2 q^{40} - 4 q^{41} + 4 q^{46} + 6 q^{49} - 8 q^{50} + 4 q^{56} - 12 q^{59}+ \cdots - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2790\mathbb{Z}\right)^\times\).

\(n\) \(1117\) \(1801\) \(2171\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −2.00000 − 1.00000i −0.894427 − 0.447214i
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 1.00000 − 2.00000i 0.316228 − 0.632456i
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 2.00000 + 1.00000i 0.447214 + 0.223607i
\(21\) 0 0
\(22\) 0 0
\(23\) − 2.00000i − 0.417029i −0.978019 − 0.208514i \(-0.933137\pi\)
0.978019 − 0.208514i \(-0.0668628\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) − 2.00000i − 0.377964i
\(29\) −10.0000 −1.85695 −0.928477 − 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 − 4.00000i 0.338062 − 0.676123i
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 0 0
\(40\) −1.00000 + 2.00000i −0.158114 + 0.316228i
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) −4.00000 + 3.00000i −0.565685 + 0.424264i
\(51\) 0 0
\(52\) − 2.00000i − 0.277350i
\(53\) − 2.00000i − 0.274721i −0.990521 − 0.137361i \(-0.956138\pi\)
0.990521 − 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) − 10.0000i − 1.31306i
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) − 1.00000i − 0.127000i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 2.00000 − 4.00000i 0.248069 − 0.496139i
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 − 0.488678i \(-0.837479\pi\)
0.872464 − 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 4.00000 + 2.00000i 0.478091 + 0.239046i
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) − 10.0000i − 1.17041i −0.810885 − 0.585206i \(-0.801014\pi\)
0.810885 − 0.585206i \(-0.198986\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) −2.00000 − 1.00000i −0.223607 − 0.111803i
\(81\) 0 0
\(82\) − 2.00000i − 0.220863i
\(83\) − 4.00000i − 0.439057i −0.975606 − 0.219529i \(-0.929548\pi\)
0.975606 − 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 2.00000i 0.208514i
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) −8.00000 − 4.00000i −0.820783 − 0.410391i
\(96\) 0 0
\(97\) 4.00000i 0.406138i 0.979164 + 0.203069i \(0.0650917\pi\)
−0.979164 + 0.203069i \(0.934908\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 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.d.d.559.2 2
3.2 odd 2 930.2.d.d.559.1 ✓ 2
5.4 even 2 inner 2790.2.d.d.559.1 2
15.2 even 4 4650.2.a.z.1.1 1
15.8 even 4 4650.2.a.u.1.1 1
15.14 odd 2 930.2.d.d.559.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.d.d.559.1 ✓ 2 3.2 odd 2
930.2.d.d.559.2 yes 2 15.14 odd 2
2790.2.d.d.559.1 2 5.4 even 2 inner
2790.2.d.d.559.2 2 1.1 even 1 trivial
4650.2.a.u.1.1 1 15.8 even 4
4650.2.a.z.1.1 1 15.2 even 4