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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2325,2,Mod(1024,2325)] 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("2325.1024"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2325.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-10,0,6,0,0,-6,0,4,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.5652184699\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 465)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1024.3
Root \(1.45161 + 1.45161i\) of defining polynomial
Character \(\chi\) \(=\) 2325.1024
Dual form 2325.2.c.l.1024.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.21432i q^{2} -1.00000i q^{3} +0.525428 q^{4} -1.21432 q^{6} -1.59210i q^{7} -3.06668i q^{8} -1.00000 q^{9} +0.622216 q^{11} -0.525428i q^{12} -0.214320i q^{13} -1.93332 q^{14} -2.67307 q^{16} +3.52543i q^{17} +1.21432i q^{18} -1.80642 q^{19} -1.59210 q^{21} -0.755569i q^{22} -6.90321i q^{23} -3.06668 q^{24} -0.260253 q^{26} +1.00000i q^{27} -0.836535i q^{28} -9.73975 q^{29} -1.00000 q^{31} -2.88739i q^{32} -0.622216i q^{33} +4.28100 q^{34} -0.525428 q^{36} -4.83654i q^{37} +2.19358i q^{38} -0.214320 q^{39} -7.47949 q^{41} +1.93332i q^{42} -8.23506i q^{43} +0.326929 q^{44} -8.38271 q^{46} +11.4652i q^{47} +2.67307i q^{48} +4.46520 q^{49} +3.52543 q^{51} -0.112610i q^{52} -13.7605i q^{53} +1.21432 q^{54} -4.88247 q^{56} +1.80642i q^{57} +11.8272i q^{58} +4.26025 q^{59} +2.85728 q^{61} +1.21432i q^{62} +1.59210i q^{63} -8.85236 q^{64} -0.755569 q^{66} -2.08097i q^{67} +1.85236i q^{68} -6.90321 q^{69} +1.31111 q^{71} +3.06668i q^{72} +1.65233i q^{73} -5.87310 q^{74} -0.949145 q^{76} -0.990632i q^{77} +0.260253i q^{78} +5.19850 q^{79} +1.00000 q^{81} +9.08250i q^{82} +5.65878i q^{83} -0.836535 q^{84} -10.0000 q^{86} +9.73975i q^{87} -1.90813i q^{88} +1.93332 q^{89} -0.341219 q^{91} -3.62714i q^{92} +1.00000i q^{93} +13.9224 q^{94} -2.88739 q^{96} +6.91750i q^{97} -5.42219i q^{98} -0.622216 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} + 6 q^{6} - 6 q^{9} + 4 q^{11} - 12 q^{14} + 10 q^{16} + 16 q^{19} + 4 q^{21} - 18 q^{24} - 28 q^{26} - 32 q^{29} - 6 q^{31} + 12 q^{34} + 10 q^{36} + 12 q^{39} + 8 q^{41} + 28 q^{44} + 16 q^{46}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2325\mathbb{Z}\right)^\times\).

\(n\) \(652\) \(776\) \(1801\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.21432i − 0.858654i −0.903149 − 0.429327i \(-0.858751\pi\)
0.903149 − 0.429327i \(-0.141249\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 0.525428 0.262714
\(5\) 0 0
\(6\) −1.21432 −0.495744
\(7\) − 1.59210i − 0.601759i −0.953662 − 0.300879i \(-0.902720\pi\)
0.953662 − 0.300879i \(-0.0972802\pi\)
\(8\) − 3.06668i − 1.08423i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0.622216 0.187605 0.0938025 − 0.995591i \(-0.470098\pi\)
0.0938025 + 0.995591i \(0.470098\pi\)
\(12\) − 0.525428i − 0.151678i
\(13\) − 0.214320i − 0.0594416i −0.999558 − 0.0297208i \(-0.990538\pi\)
0.999558 − 0.0297208i \(-0.00946182\pi\)
\(14\) −1.93332 −0.516702
\(15\) 0 0
\(16\) −2.67307 −0.668268
\(17\) 3.52543i 0.855042i 0.904005 + 0.427521i \(0.140613\pi\)
−0.904005 + 0.427521i \(0.859387\pi\)
\(18\) 1.21432i 0.286218i
\(19\) −1.80642 −0.414422 −0.207211 − 0.978296i \(-0.566439\pi\)
−0.207211 + 0.978296i \(0.566439\pi\)
\(20\) 0 0
\(21\) −1.59210 −0.347426
\(22\) − 0.755569i − 0.161088i
\(23\) − 6.90321i − 1.43942i −0.694275 − 0.719710i \(-0.744275\pi\)
0.694275 − 0.719710i \(-0.255725\pi\)
\(24\) −3.06668 −0.625983
\(25\) 0 0
\(26\) −0.260253 −0.0510398
\(27\) 1.00000i 0.192450i
\(28\) − 0.836535i − 0.158090i
\(29\) −9.73975 −1.80863 −0.904313 − 0.426870i \(-0.859616\pi\)
−0.904313 + 0.426870i \(0.859616\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) − 2.88739i − 0.510423i
\(33\) − 0.622216i − 0.108314i
\(34\) 4.28100 0.734185
\(35\) 0 0
\(36\) −0.525428 −0.0875713
\(37\) − 4.83654i − 0.795122i −0.917576 − 0.397561i \(-0.869857\pi\)
0.917576 − 0.397561i \(-0.130143\pi\)
\(38\) 2.19358i 0.355845i
\(39\) −0.214320 −0.0343186
\(40\) 0 0
\(41\) −7.47949 −1.16810 −0.584050 − 0.811717i \(-0.698533\pi\)
−0.584050 + 0.811717i \(0.698533\pi\)
\(42\) 1.93332i 0.298318i
\(43\) − 8.23506i − 1.25584i −0.778280 − 0.627918i \(-0.783907\pi\)
0.778280 − 0.627918i \(-0.216093\pi\)
\(44\) 0.326929 0.0492864
\(45\) 0 0
\(46\) −8.38271 −1.23596
\(47\) 11.4652i 1.67237i 0.548446 + 0.836186i \(0.315220\pi\)
−0.548446 + 0.836186i \(0.684780\pi\)
\(48\) 2.67307i 0.385825i
\(49\) 4.46520 0.637886
\(50\) 0 0
\(51\) 3.52543 0.493659
\(52\) − 0.112610i − 0.0156161i
\(53\) − 13.7605i − 1.89015i −0.326855 − 0.945074i \(-0.605989\pi\)
0.326855 − 0.945074i \(-0.394011\pi\)
\(54\) 1.21432 0.165248
\(55\) 0 0
\(56\) −4.88247 −0.652447
\(57\) 1.80642i 0.239267i
\(58\) 11.8272i 1.55298i
\(59\) 4.26025 0.554638 0.277319 − 0.960778i \(-0.410554\pi\)
0.277319 + 0.960778i \(0.410554\pi\)
\(60\) 0 0
\(61\) 2.85728 0.365837 0.182919 − 0.983128i \(-0.441446\pi\)
0.182919 + 0.983128i \(0.441446\pi\)
\(62\) 1.21432i 0.154219i
\(63\) 1.59210i 0.200586i
\(64\) −8.85236 −1.10654
\(65\) 0 0
\(66\) −0.755569 −0.0930041
\(67\) − 2.08097i − 0.254231i −0.991888 − 0.127115i \(-0.959428\pi\)
0.991888 − 0.127115i \(-0.0405718\pi\)
\(68\) 1.85236i 0.224631i
\(69\) −6.90321 −0.831049
\(70\) 0 0
\(71\) 1.31111 0.155600 0.0777999 − 0.996969i \(-0.475210\pi\)
0.0777999 + 0.996969i \(0.475210\pi\)
\(72\) 3.06668i 0.361411i
\(73\) 1.65233i 0.193390i 0.995314 + 0.0966951i \(0.0308272\pi\)
−0.995314 + 0.0966951i \(0.969173\pi\)
\(74\) −5.87310 −0.682734
\(75\) 0 0
\(76\) −0.949145 −0.108874
\(77\) − 0.990632i − 0.112893i
\(78\) 0.260253i 0.0294678i
\(79\) 5.19850 0.584877 0.292438 − 0.956284i \(-0.405533\pi\)
0.292438 + 0.956284i \(0.405533\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 9.08250i 1.00299i
\(83\) 5.65878i 0.621132i 0.950552 + 0.310566i \(0.100519\pi\)
−0.950552 + 0.310566i \(0.899481\pi\)
\(84\) −0.836535 −0.0912735
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) 9.73975i 1.04421i
\(88\) − 1.90813i − 0.203408i
\(89\) 1.93332 0.204932 0.102466 − 0.994737i \(-0.467327\pi\)
0.102466 + 0.994737i \(0.467327\pi\)
\(90\) 0 0
\(91\) −0.341219 −0.0357695
\(92\) − 3.62714i − 0.378155i
\(93\) 1.00000i 0.103695i
\(94\) 13.9224 1.43599
\(95\) 0 0
\(96\) −2.88739 −0.294693
\(97\) 6.91750i 0.702366i 0.936307 + 0.351183i \(0.114220\pi\)
−0.936307 + 0.351183i \(0.885780\pi\)
\(98\) − 5.42219i − 0.547723i
\(99\) −0.622216 −0.0625350
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 2325.2.c.l.1024.3 6
5.2 odd 4 2325.2.a.p.1.3 3
5.3 odd 4 465.2.a.g.1.1 ✓ 3
5.4 even 2 inner 2325.2.c.l.1024.4 6
15.2 even 4 6975.2.a.bi.1.1 3
15.8 even 4 1395.2.a.h.1.3 3
20.3 even 4 7440.2.a.bm.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.g.1.1 ✓ 3 5.3 odd 4
1395.2.a.h.1.3 3 15.8 even 4
2325.2.a.p.1.3 3 5.2 odd 4
2325.2.c.l.1024.3 6 1.1 even 1 trivial
2325.2.c.l.1024.4 6 5.4 even 2 inner
6975.2.a.bi.1.1 3 15.2 even 4
7440.2.a.bm.1.3 3 20.3 even 4