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(1,2325)] 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("2325.1"); 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, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-3,-3,5,0,3,-2,-9,3,0,2] 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: yes
Analytic conductor: \(18.5652184699\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 465)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 2325.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.21432 q^{2} -1.00000 q^{3} -0.525428 q^{4} -1.21432 q^{6} +1.59210 q^{7} -3.06668 q^{8} +1.00000 q^{9} +0.622216 q^{11} +0.525428 q^{12} -0.214320 q^{13} +1.93332 q^{14} -2.67307 q^{16} -3.52543 q^{17} +1.21432 q^{18} +1.80642 q^{19} -1.59210 q^{21} +0.755569 q^{22} -6.90321 q^{23} +3.06668 q^{24} -0.260253 q^{26} -1.00000 q^{27} -0.836535 q^{28} +9.73975 q^{29} -1.00000 q^{31} +2.88739 q^{32} -0.622216 q^{33} -4.28100 q^{34} -0.525428 q^{36} +4.83654 q^{37} +2.19358 q^{38} +0.214320 q^{39} -7.47949 q^{41} -1.93332 q^{42} -8.23506 q^{43} -0.326929 q^{44} -8.38271 q^{46} -11.4652 q^{47} +2.67307 q^{48} -4.46520 q^{49} +3.52543 q^{51} +0.112610 q^{52} -13.7605 q^{53} -1.21432 q^{54} -4.88247 q^{56} -1.80642 q^{57} +11.8272 q^{58} -4.26025 q^{59} +2.85728 q^{61} -1.21432 q^{62} +1.59210 q^{63} +8.85236 q^{64} -0.755569 q^{66} +2.08097 q^{67} +1.85236 q^{68} +6.90321 q^{69} +1.31111 q^{71} -3.06668 q^{72} +1.65233 q^{73} +5.87310 q^{74} -0.949145 q^{76} +0.990632 q^{77} +0.260253 q^{78} -5.19850 q^{79} +1.00000 q^{81} -9.08250 q^{82} +5.65878 q^{83} +0.836535 q^{84} -10.0000 q^{86} -9.73975 q^{87} -1.90813 q^{88} -1.93332 q^{89} -0.341219 q^{91} +3.62714 q^{92} +1.00000 q^{93} -13.9224 q^{94} -2.88739 q^{96} -6.91750 q^{97} -5.42219 q^{98} +0.622216 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 3 q^{3} + 5 q^{4} + 3 q^{6} - 2 q^{7} - 9 q^{8} + 3 q^{9} + 2 q^{11} - 5 q^{12} + 6 q^{13} + 6 q^{14} + 5 q^{16} - 4 q^{17} - 3 q^{18} - 8 q^{19} + 2 q^{21} + 2 q^{22} - 14 q^{23} + 9 q^{24}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.21432 0.858654 0.429327 − 0.903149i \(-0.358751\pi\)
0.429327 + 0.903149i \(0.358751\pi\)
\(3\) −1.00000 −0.577350
\(4\) −0.525428 −0.262714
\(5\) 0 0
\(6\) −1.21432 −0.495744
\(7\) 1.59210 0.601759 0.300879 − 0.953662i \(-0.402720\pi\)
0.300879 + 0.953662i \(0.402720\pi\)
\(8\) −3.06668 −1.08423
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.622216 0.187605 0.0938025 − 0.995591i \(-0.470098\pi\)
0.0938025 + 0.995591i \(0.470098\pi\)
\(12\) 0.525428 0.151678
\(13\) −0.214320 −0.0594416 −0.0297208 − 0.999558i \(-0.509462\pi\)
−0.0297208 + 0.999558i \(0.509462\pi\)
\(14\) 1.93332 0.516702
\(15\) 0 0
\(16\) −2.67307 −0.668268
\(17\) −3.52543 −0.855042 −0.427521 − 0.904005i \(-0.640613\pi\)
−0.427521 + 0.904005i \(0.640613\pi\)
\(18\) 1.21432 0.286218
\(19\) 1.80642 0.414422 0.207211 − 0.978296i \(-0.433561\pi\)
0.207211 + 0.978296i \(0.433561\pi\)
\(20\) 0 0
\(21\) −1.59210 −0.347426
\(22\) 0.755569 0.161088
\(23\) −6.90321 −1.43942 −0.719710 − 0.694275i \(-0.755725\pi\)
−0.719710 + 0.694275i \(0.755725\pi\)
\(24\) 3.06668 0.625983
\(25\) 0 0
\(26\) −0.260253 −0.0510398
\(27\) −1.00000 −0.192450
\(28\) −0.836535 −0.158090
\(29\) 9.73975 1.80863 0.904313 − 0.426870i \(-0.140384\pi\)
0.904313 + 0.426870i \(0.140384\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 2.88739 0.510423
\(33\) −0.622216 −0.108314
\(34\) −4.28100 −0.734185
\(35\) 0 0
\(36\) −0.525428 −0.0875713
\(37\) 4.83654 0.795122 0.397561 − 0.917576i \(-0.369857\pi\)
0.397561 + 0.917576i \(0.369857\pi\)
\(38\) 2.19358 0.355845
\(39\) 0.214320 0.0343186
\(40\) 0 0
\(41\) −7.47949 −1.16810 −0.584050 − 0.811717i \(-0.698533\pi\)
−0.584050 + 0.811717i \(0.698533\pi\)
\(42\) −1.93332 −0.298318
\(43\) −8.23506 −1.25584 −0.627918 − 0.778280i \(-0.716093\pi\)
−0.627918 + 0.778280i \(0.716093\pi\)
\(44\) −0.326929 −0.0492864
\(45\) 0 0
\(46\) −8.38271 −1.23596
\(47\) −11.4652 −1.67237 −0.836186 − 0.548446i \(-0.815220\pi\)
−0.836186 + 0.548446i \(0.815220\pi\)
\(48\) 2.67307 0.385825
\(49\) −4.46520 −0.637886
\(50\) 0 0
\(51\) 3.52543 0.493659
\(52\) 0.112610 0.0156161
\(53\) −13.7605 −1.89015 −0.945074 − 0.326855i \(-0.894011\pi\)
−0.945074 + 0.326855i \(0.894011\pi\)
\(54\) −1.21432 −0.165248
\(55\) 0 0
\(56\) −4.88247 −0.652447
\(57\) −1.80642 −0.239267
\(58\) 11.8272 1.55298
\(59\) −4.26025 −0.554638 −0.277319 − 0.960778i \(-0.589446\pi\)
−0.277319 + 0.960778i \(0.589446\pi\)
\(60\) 0 0
\(61\) 2.85728 0.365837 0.182919 − 0.983128i \(-0.441446\pi\)
0.182919 + 0.983128i \(0.441446\pi\)
\(62\) −1.21432 −0.154219
\(63\) 1.59210 0.200586
\(64\) 8.85236 1.10654
\(65\) 0 0
\(66\) −0.755569 −0.0930041
\(67\) 2.08097 0.254231 0.127115 − 0.991888i \(-0.459428\pi\)
0.127115 + 0.991888i \(0.459428\pi\)
\(68\) 1.85236 0.224631
\(69\) 6.90321 0.831049
\(70\) 0 0
\(71\) 1.31111 0.155600 0.0777999 − 0.996969i \(-0.475210\pi\)
0.0777999 + 0.996969i \(0.475210\pi\)
\(72\) −3.06668 −0.361411
\(73\) 1.65233 0.193390 0.0966951 − 0.995314i \(-0.469173\pi\)
0.0966951 + 0.995314i \(0.469173\pi\)
\(74\) 5.87310 0.682734
\(75\) 0 0
\(76\) −0.949145 −0.108874
\(77\) 0.990632 0.112893
\(78\) 0.260253 0.0294678
\(79\) −5.19850 −0.584877 −0.292438 − 0.956284i \(-0.594467\pi\)
−0.292438 + 0.956284i \(0.594467\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −9.08250 −1.00299
\(83\) 5.65878 0.621132 0.310566 − 0.950552i \(-0.399481\pi\)
0.310566 + 0.950552i \(0.399481\pi\)
\(84\) 0.836535 0.0912735
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) −9.73975 −1.04421
\(88\) −1.90813 −0.203408
\(89\) −1.93332 −0.204932 −0.102466 − 0.994737i \(-0.532673\pi\)
−0.102466 + 0.994737i \(0.532673\pi\)
\(90\) 0 0
\(91\) −0.341219 −0.0357695
\(92\) 3.62714 0.378155
\(93\) 1.00000 0.103695
\(94\) −13.9224 −1.43599
\(95\) 0 0
\(96\) −2.88739 −0.294693
\(97\) −6.91750 −0.702366 −0.351183 − 0.936307i \(-0.614220\pi\)
−0.351183 + 0.936307i \(0.614220\pi\)
\(98\) −5.42219 −0.547723
\(99\) 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.a.p.1.3 3
3.2 odd 2 6975.2.a.bi.1.1 3
5.2 odd 4 2325.2.c.l.1024.4 6
5.3 odd 4 2325.2.c.l.1024.3 6
5.4 even 2 465.2.a.g.1.1 ✓ 3
15.14 odd 2 1395.2.a.h.1.3 3
20.19 odd 2 7440.2.a.bm.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.g.1.1 ✓ 3 5.4 even 2
1395.2.a.h.1.3 3 15.14 odd 2
2325.2.a.p.1.3 3 1.1 even 1 trivial
2325.2.c.l.1024.3 6 5.3 odd 4
2325.2.c.l.1024.4 6 5.2 odd 4
6975.2.a.bi.1.1 3 3.2 odd 2
7440.2.a.bm.1.3 3 20.19 odd 2