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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4650,2,Mod(3349,4650)] 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("4650.3349"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,0,2,0,0,-2,0,-8,0,0,0,0,2,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(37.1304369399\)
Analytic rank: \(0\)
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 3349.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4650.3349
Dual form 4650.2.d.o.3349.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} -4.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} -6.00000 q^{26} +1.00000i q^{27} -2.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} +6.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} +4.00000 q^{44} +4.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} -2.00000i q^{58} +4.00000 q^{59} -6.00000 q^{61} -1.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} -16.0000i q^{67} +2.00000i q^{68} -4.00000 q^{69} -12.0000 q^{71} +1.00000i q^{72} -6.00000i q^{73} -2.00000 q^{74} -4.00000 q^{76} +6.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -12.0000i q^{83} +4.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} +18.0000 q^{89} +4.00000i q^{92} +1.00000i q^{93} +1.00000 q^{96} +14.0000i q^{97} +7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{16} + 8 q^{19} - 2 q^{24} - 12 q^{26} - 4 q^{29} - 2 q^{31} + 4 q^{34} + 2 q^{36} + 12 q^{39} - 12 q^{41} + 8 q^{44} + 8 q^{46} + 14 q^{49} - 4 q^{51}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times\).

\(n\) \(1801\) \(2977\) \(3101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 − 0.242536i \(-0.922021\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 4.00000i − 0.852803i
\(23\) − 4.00000i − 0.834058i −0.908893 − 0.417029i \(-0.863071\pi\)
0.908893 − 0.417029i \(-0.136929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −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
\(33\) 4.00000i 0.696311i
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) − 6.00000i − 0.832050i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) − 2.00000i − 0.262613i
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) − 1.00000i − 0.127000i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) − 16.0000i − 1.95471i −0.211604 − 0.977356i \(-0.567869\pi\)
0.211604 − 0.977356i \(-0.432131\pi\)
\(68\) 2.00000i 0.242536i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 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
\(77\) 0 0
\(78\) 6.00000i 0.679366i
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 6.00000i − 0.662589i
\(83\) − 12.0000i − 1.31717i −0.752506 − 0.658586i \(-0.771155\pi\)
0.752506 − 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 2.00000i 0.214423i
\(88\) 4.00000i 0.426401i
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000i 0.417029i
\(93\) 1.00000i 0.103695i
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 7.00000i 0.707107i
\(99\) 4.00000 0.402015
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 4650.2.d.o.3349.2 2
5.2 odd 4 930.2.a.b.1.1 ✓ 1
5.3 odd 4 4650.2.a.bp.1.1 1
5.4 even 2 inner 4650.2.d.o.3349.1 2
15.2 even 4 2790.2.a.ba.1.1 1
20.7 even 4 7440.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.b.1.1 ✓ 1 5.2 odd 4
2790.2.a.ba.1.1 1 15.2 even 4
4650.2.a.bp.1.1 1 5.3 odd 4
4650.2.d.o.3349.1 2 5.4 even 2 inner
4650.2.d.o.3349.2 2 1.1 even 1 trivial
7440.2.a.q.1.1 1 20.7 even 4