Show commands: Magma / Pari/GP / SageMath

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 1024.5
Root \(-0.854638 - 0.854638i\) of defining polynomial
Character \(\chi\) \(=\) 2325.1024
Dual form 2325.2.c.l.1024.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.53919i q^{2} -1.00000i q^{3} -0.369102 q^{4} +1.53919 q^{6} +4.87936i q^{7} +2.51026i q^{8} -1.00000 q^{9} +4.34017 q^{11} +0.369102i q^{12} +2.53919i q^{13} -7.51026 q^{14} -4.60197 q^{16} +2.63090i q^{17} -1.53919i q^{18} +7.41855 q^{19} +4.87936 q^{21} +6.68035i q^{22} -2.29072i q^{23} +2.51026 q^{24} -3.90829 q^{26} +1.00000i q^{27} -1.80098i q^{28} -6.09171 q^{29} -1.00000 q^{31} -2.06278i q^{32} -4.34017i q^{33} -4.04945 q^{34} +0.369102 q^{36} -5.80098i q^{37} +11.4186i q^{38} +2.53919 q^{39} -0.183417 q^{41} +7.51026i q^{42} +6.49693i q^{43} -1.60197 q^{44} +3.52586 q^{46} -9.80817i q^{47} +4.60197i q^{48} -16.8082 q^{49} +2.63090 q^{51} -0.937221i q^{52} +1.86603i q^{53} -1.53919 q^{54} -12.2485 q^{56} -7.41855i q^{57} -9.37629i q^{58} +7.90829 q^{59} -8.15676 q^{61} -1.53919i q^{62} -4.87936i q^{63} -6.02893 q^{64} +6.68035 q^{66} -10.4813i q^{67} -0.971071i q^{68} -2.29072 q^{69} +3.17009 q^{71} -2.51026i q^{72} +15.5597i q^{73} +8.92881 q^{74} -2.73820 q^{76} +21.1773i q^{77} +3.90829i q^{78} +6.23287 q^{79} +1.00000 q^{81} -0.282314i q^{82} -6.38962i q^{83} -1.80098 q^{84} -10.0000 q^{86} +6.09171i q^{87} +10.8950i q^{88} +7.51026 q^{89} -12.3896 q^{91} +0.845512i q^{92} +1.00000i q^{93} +15.0966 q^{94} -2.06278 q^{96} +16.2823i q^{97} -25.8710i q^{98} -4.34017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} + 6 q^{6} - 6 q^{9} + 4 q^{11} - 12 q^{14} + 10 q^{16} + 16 q^{19} + 4 q^{21} - 18 q^{24} - 28 q^{26} - 32 q^{29} - 6 q^{31} + 12 q^{34} + 10 q^{36} + 12 q^{39} + 8 q^{41} + 28 q^{44} + 16 q^{46}+ \cdots - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.53919i 1.08837i 0.838965 + 0.544185i \(0.183161\pi\)
−0.838965 + 0.544185i \(0.816839\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −0.369102 −0.184551
\(5\) 0 0
\(6\) 1.53919 0.628371
\(7\) 4.87936i 1.84423i 0.386921 + 0.922113i \(0.373538\pi\)
−0.386921 + 0.922113i \(0.626462\pi\)
\(8\) 2.51026i 0.887511i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.34017 1.30861 0.654306 − 0.756230i \(-0.272961\pi\)
0.654306 + 0.756230i \(0.272961\pi\)
\(12\) 0.369102i 0.106551i
\(13\) 2.53919i 0.704244i 0.935954 + 0.352122i \(0.114540\pi\)
−0.935954 + 0.352122i \(0.885460\pi\)
\(14\) −7.51026 −2.00720
\(15\) 0 0
\(16\) −4.60197 −1.15049
\(17\) 2.63090i 0.638086i 0.947740 + 0.319043i \(0.103362\pi\)
−0.947740 + 0.319043i \(0.896638\pi\)
\(18\) − 1.53919i − 0.362790i
\(19\) 7.41855 1.70193 0.850966 − 0.525221i \(-0.176017\pi\)
0.850966 + 0.525221i \(0.176017\pi\)
\(20\) 0 0
\(21\) 4.87936 1.06476
\(22\) 6.68035i 1.42425i
\(23\) − 2.29072i − 0.477649i −0.971063 − 0.238825i \(-0.923238\pi\)
0.971063 − 0.238825i \(-0.0767621\pi\)
\(24\) 2.51026 0.512405
\(25\) 0 0
\(26\) −3.90829 −0.766479
\(27\) 1.00000i 0.192450i
\(28\) − 1.80098i − 0.340354i
\(29\) −6.09171 −1.13120 −0.565601 − 0.824679i \(-0.691356\pi\)
−0.565601 + 0.824679i \(0.691356\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) − 2.06278i − 0.364651i
\(33\) − 4.34017i − 0.755527i
\(34\) −4.04945 −0.694475
\(35\) 0 0
\(36\) 0.369102 0.0615171
\(37\) − 5.80098i − 0.953676i −0.878991 − 0.476838i \(-0.841783\pi\)
0.878991 − 0.476838i \(-0.158217\pi\)
\(38\) 11.4186i 1.85233i
\(39\) 2.53919 0.406596
\(40\) 0 0
\(41\) −0.183417 −0.0286450 −0.0143225 − 0.999897i \(-0.504559\pi\)
−0.0143225 + 0.999897i \(0.504559\pi\)
\(42\) 7.51026i 1.15886i
\(43\) 6.49693i 0.990772i 0.868673 + 0.495386i \(0.164973\pi\)
−0.868673 + 0.495386i \(0.835027\pi\)
\(44\) −1.60197 −0.241506
\(45\) 0 0
\(46\) 3.52586 0.519859
\(47\) − 9.80817i − 1.43067i −0.698782 − 0.715334i \(-0.746274\pi\)
0.698782 − 0.715334i \(-0.253726\pi\)
\(48\) 4.60197i 0.664237i
\(49\) −16.8082 −2.40117
\(50\) 0 0
\(51\) 2.63090 0.368399
\(52\) − 0.937221i − 0.129969i
\(53\) 1.86603i 0.256319i 0.991754 + 0.128160i \(0.0409070\pi\)
−0.991754 + 0.128160i \(0.959093\pi\)
\(54\) −1.53919 −0.209457
\(55\) 0 0
\(56\) −12.2485 −1.63677
\(57\) − 7.41855i − 0.982611i
\(58\) − 9.37629i − 1.23117i
\(59\) 7.90829 1.02957 0.514786 − 0.857319i \(-0.327871\pi\)
0.514786 + 0.857319i \(0.327871\pi\)
\(60\) 0 0
\(61\) −8.15676 −1.04437 −0.522183 − 0.852834i \(-0.674882\pi\)
−0.522183 + 0.852834i \(0.674882\pi\)
\(62\) − 1.53919i − 0.195477i
\(63\) − 4.87936i − 0.614742i
\(64\) −6.02893 −0.753616
\(65\) 0 0
\(66\) 6.68035 0.822294
\(67\) − 10.4813i − 1.28050i −0.768167 − 0.640249i \(-0.778831\pi\)
0.768167 − 0.640249i \(-0.221169\pi\)
\(68\) − 0.971071i − 0.117760i
\(69\) −2.29072 −0.275771
\(70\) 0 0
\(71\) 3.17009 0.376220 0.188110 − 0.982148i \(-0.439764\pi\)
0.188110 + 0.982148i \(0.439764\pi\)
\(72\) − 2.51026i − 0.295837i
\(73\) 15.5597i 1.82113i 0.413370 + 0.910563i \(0.364352\pi\)
−0.413370 + 0.910563i \(0.635648\pi\)
\(74\) 8.92881 1.03795
\(75\) 0 0
\(76\) −2.73820 −0.314094
\(77\) 21.1773i 2.41337i
\(78\) 3.90829i 0.442527i
\(79\) 6.23287 0.701252 0.350626 − 0.936516i \(-0.385969\pi\)
0.350626 + 0.936516i \(0.385969\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 0.282314i − 0.0311764i
\(83\) − 6.38962i − 0.701352i −0.936497 − 0.350676i \(-0.885952\pi\)
0.936497 − 0.350676i \(-0.114048\pi\)
\(84\) −1.80098 −0.196503
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) 6.09171i 0.653100i
\(88\) 10.8950i 1.16141i
\(89\) 7.51026 0.796086 0.398043 − 0.917367i \(-0.369689\pi\)
0.398043 + 0.917367i \(0.369689\pi\)
\(90\) 0 0
\(91\) −12.3896 −1.29879
\(92\) 0.845512i 0.0881507i
\(93\) 1.00000i 0.103695i
\(94\) 15.0966 1.55710
\(95\) 0 0
\(96\) −2.06278 −0.210532
\(97\) 16.2823i 1.65322i 0.562776 + 0.826609i \(0.309733\pi\)
−0.562776 + 0.826609i \(0.690267\pi\)
\(98\) − 25.8710i − 2.61336i
\(99\) −4.34017 −0.436204
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.1024.5 6
5.2 odd 4 2325.2.a.p.1.2 3
5.3 odd 4 465.2.a.g.1.2 ✓ 3
5.4 even 2 inner 2325.2.c.l.1024.2 6
15.2 even 4 6975.2.a.bi.1.2 3
15.8 even 4 1395.2.a.h.1.2 3
20.3 even 4 7440.2.a.bm.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.g.1.2 ✓ 3 5.3 odd 4
1395.2.a.h.1.2 3 15.8 even 4
2325.2.a.p.1.2 3 5.2 odd 4
2325.2.c.l.1024.2 6 5.4 even 2 inner
2325.2.c.l.1024.5 6 1.1 even 1 trivial
6975.2.a.bi.1.2 3 15.2 even 4
7440.2.a.bm.1.1 3 20.3 even 4