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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,-8] 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: yes
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.a.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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +2.00000i q^{7} -1.00000i q^{8} +(-1.00000 - 2.00000i) q^{10} -4.00000 q^{11} +2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -4.00000 q^{19} +(2.00000 - 1.00000i) q^{20} -4.00000i q^{22} +(3.00000 - 4.00000i) q^{25} -2.00000 q^{26} -2.00000i q^{28} -2.00000 q^{29} +1.00000 q^{31} +1.00000i q^{32} -6.00000 q^{34} +(-2.00000 - 4.00000i) q^{35} -2.00000i q^{37} -4.00000i q^{38} +(1.00000 + 2.00000i) q^{40} +4.00000i q^{43} +4.00000 q^{44} +3.00000 q^{49} +(4.00000 + 3.00000i) q^{50} -2.00000i q^{52} -2.00000i q^{53} +(8.00000 - 4.00000i) q^{55} +2.00000 q^{56} -2.00000i q^{58} -14.0000 q^{59} +10.0000 q^{61} +1.00000i q^{62} -1.00000 q^{64} +(-2.00000 - 4.00000i) q^{65} -2.00000i q^{67} -6.00000i q^{68} +(4.00000 - 2.00000i) q^{70} -6.00000 q^{71} -6.00000i q^{73} +2.00000 q^{74} +4.00000 q^{76} -8.00000i q^{77} +4.00000 q^{79} +(-2.00000 + 1.00000i) q^{80} -12.0000i q^{83} +(-6.00000 - 12.0000i) q^{85} -4.00000 q^{86} +4.00000i q^{88} -6.00000 q^{89} -4.00000 q^{91} +(8.00000 - 4.00000i) q^{95} -12.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 4 q^{5} - 2 q^{10} - 8 q^{11} - 4 q^{14} + 2 q^{16} - 8 q^{19} + 4 q^{20} + 6 q^{25} - 4 q^{26} - 4 q^{29} + 2 q^{31} - 12 q^{34} - 4 q^{35} + 2 q^{40} + 8 q^{44} + 6 q^{49} + 8 q^{50} + 16 q^{55}+ \cdots + 16 q^{95}+O(q^{100}) \) 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
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 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
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 2.00000 − 1.00000i 0.447214 − 0.223607i
\(21\) 0 0
\(22\) − 4.00000i − 0.852803i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 3.00000 − 4.00000i 0.600000 − 0.800000i
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) − 2.00000i − 0.377964i
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) −2.00000 − 4.00000i −0.338062 − 0.676123i
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) 0 0
\(40\) 1.00000 + 2.00000i 0.158114 + 0.316228i
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 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
\(55\) 8.00000 − 4.00000i 1.07872 − 0.539360i
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) − 2.00000i − 0.262613i
\(59\) −14.0000 −1.82264 −0.911322 − 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 1.00000i 0.127000i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −2.00000 − 4.00000i −0.248069 − 0.496139i
\(66\) 0 0
\(67\) − 2.00000i − 0.244339i −0.992509 − 0.122169i \(-0.961015\pi\)
0.992509 − 0.122169i \(-0.0389851\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) 0 0
\(70\) 4.00000 − 2.00000i 0.478091 − 0.239046i
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) − 6.00000i − 0.702247i −0.936329 − 0.351123i \(-0.885800\pi\)
0.936329 − 0.351123i \(-0.114200\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) − 8.00000i − 0.911685i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −2.00000 + 1.00000i −0.223607 + 0.111803i
\(81\) 0 0
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 − 0.658586i \(-0.771155\pi\)
0.752506 − 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) −6.00000 − 12.0000i −0.650791 − 1.30158i
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 4.00000i 0.426401i
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 − 4.00000i 0.820783 − 0.410391i
\(96\) 0 0
\(97\) − 12.0000i − 1.21842i −0.793011 − 0.609208i \(-0.791488\pi\)
0.793011 − 0.609208i \(-0.208512\pi\)
\(98\) 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.a.559.2 yes 2
3.2 odd 2 2790.2.d.h.559.1 yes 2
5.4 even 2 inner 2790.2.d.a.559.1 ✓ 2
15.14 odd 2 2790.2.d.h.559.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2790.2.d.a.559.1 ✓ 2 5.4 even 2 inner
2790.2.d.a.559.2 yes 2 1.1 even 1 trivial
2790.2.d.h.559.1 yes 2 3.2 odd 2
2790.2.d.h.559.2 yes 2 15.14 odd 2