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 1.1
Root \(-1.48119\) 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)
 
\(f(q)\) \(=\) \(q-2.67513 q^{2} -1.00000 q^{3} +5.15633 q^{4} +2.67513 q^{6} +1.28726 q^{7} -8.44358 q^{8} +1.00000 q^{9} -2.96239 q^{11} -5.15633 q^{12} +3.67513 q^{13} -3.44358 q^{14} +12.2750 q^{16} +2.15633 q^{17} -2.67513 q^{18} -2.38787 q^{19} -1.28726 q^{21} +7.92478 q^{22} -4.80606 q^{23} +8.44358 q^{24} -9.83146 q^{26} -1.00000 q^{27} +6.63752 q^{28} +0.168544 q^{29} -1.00000 q^{31} -15.9502 q^{32} +2.96239 q^{33} -5.76845 q^{34} +5.15633 q^{36} -2.63752 q^{37} +6.38787 q^{38} -3.67513 q^{39} +11.6629 q^{41} +3.44358 q^{42} +3.73813 q^{43} -15.2750 q^{44} +12.8568 q^{46} -12.3430 q^{47} -12.2750 q^{48} -5.34297 q^{49} -2.15633 q^{51} +18.9502 q^{52} +3.89446 q^{53} +2.67513 q^{54} -10.8691 q^{56} +2.38787 q^{57} -0.450877 q^{58} -13.8315 q^{59} -12.7005 q^{61} +2.67513 q^{62} +1.28726 q^{63} +18.1187 q^{64} -7.92478 q^{66} -12.5623 q^{67} +11.1187 q^{68} +4.80606 q^{69} -0.481194 q^{71} -8.44358 q^{72} -5.21203 q^{73} +7.05571 q^{74} -12.3127 q^{76} -3.81336 q^{77} +9.83146 q^{78} +15.4314 q^{79} +1.00000 q^{81} -31.1998 q^{82} +10.7308 q^{83} -6.63752 q^{84} -10.0000 q^{86} -0.168544 q^{87} +25.0132 q^{88} +3.44358 q^{89} +4.73084 q^{91} -24.7816 q^{92} +1.00000 q^{93} +33.0191 q^{94} +15.9502 q^{96} +15.1998 q^{97} +14.2931 q^{98} -2.96239 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 3 q^{3} + 5 q^{4} + 3 q^{6} - 2 q^{7} - 9 q^{8} + 3 q^{9} + 2 q^{11} - 5 q^{12} + 6 q^{13} + 6 q^{14} + 5 q^{16} - 4 q^{17} - 3 q^{18} - 8 q^{19} + 2 q^{21} + 2 q^{22} - 14 q^{23} + 9 q^{24}+ \cdots + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.67513 −1.89160 −0.945802 − 0.324745i \(-0.894721\pi\)
−0.945802 + 0.324745i \(0.894721\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.15633 2.57816
\(5\) 0 0
\(6\) 2.67513 1.09212
\(7\) 1.28726 0.486538 0.243269 − 0.969959i \(-0.421780\pi\)
0.243269 + 0.969959i \(0.421780\pi\)
\(8\) −8.44358 −2.98526
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.96239 −0.893194 −0.446597 − 0.894735i \(-0.647364\pi\)
−0.446597 + 0.894735i \(0.647364\pi\)
\(12\) −5.15633 −1.48850
\(13\) 3.67513 1.01930 0.509649 − 0.860382i \(-0.329775\pi\)
0.509649 + 0.860382i \(0.329775\pi\)
\(14\) −3.44358 −0.920336
\(15\) 0 0
\(16\) 12.2750 3.06876
\(17\) 2.15633 0.522986 0.261493 − 0.965205i \(-0.415785\pi\)
0.261493 + 0.965205i \(0.415785\pi\)
\(18\) −2.67513 −0.630534
\(19\) −2.38787 −0.547816 −0.273908 − 0.961756i \(-0.588316\pi\)
−0.273908 + 0.961756i \(0.588316\pi\)
\(20\) 0 0
\(21\) −1.28726 −0.280903
\(22\) 7.92478 1.68957
\(23\) −4.80606 −1.00213 −0.501067 − 0.865409i \(-0.667059\pi\)
−0.501067 + 0.865409i \(0.667059\pi\)
\(24\) 8.44358 1.72354
\(25\) 0 0
\(26\) −9.83146 −1.92811
\(27\) −1.00000 −0.192450
\(28\) 6.63752 1.25437
\(29\) 0.168544 0.0312978 0.0156489 − 0.999878i \(-0.495019\pi\)
0.0156489 + 0.999878i \(0.495019\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) −15.9502 −2.81962
\(33\) 2.96239 0.515686
\(34\) −5.76845 −0.989281
\(35\) 0 0
\(36\) 5.15633 0.859388
\(37\) −2.63752 −0.433606 −0.216803 − 0.976215i \(-0.569563\pi\)
−0.216803 + 0.976215i \(0.569563\pi\)
\(38\) 6.38787 1.03625
\(39\) −3.67513 −0.588492
\(40\) 0 0
\(41\) 11.6629 1.82144 0.910720 − 0.413023i \(-0.135527\pi\)
0.910720 + 0.413023i \(0.135527\pi\)
\(42\) 3.44358 0.531356
\(43\) 3.73813 0.570060 0.285030 − 0.958519i \(-0.407996\pi\)
0.285030 + 0.958519i \(0.407996\pi\)
\(44\) −15.2750 −2.30280
\(45\) 0 0
\(46\) 12.8568 1.89564
\(47\) −12.3430 −1.80041 −0.900203 − 0.435470i \(-0.856582\pi\)
−0.900203 + 0.435470i \(0.856582\pi\)
\(48\) −12.2750 −1.77175
\(49\) −5.34297 −0.763281
\(50\) 0 0
\(51\) −2.15633 −0.301946
\(52\) 18.9502 2.62792
\(53\) 3.89446 0.534945 0.267473 − 0.963565i \(-0.413812\pi\)
0.267473 + 0.963565i \(0.413812\pi\)
\(54\) 2.67513 0.364039
\(55\) 0 0
\(56\) −10.8691 −1.45244
\(57\) 2.38787 0.316282
\(58\) −0.450877 −0.0592031
\(59\) −13.8315 −1.80070 −0.900351 − 0.435164i \(-0.856690\pi\)
−0.900351 + 0.435164i \(0.856690\pi\)
\(60\) 0 0
\(61\) −12.7005 −1.62614 −0.813068 − 0.582169i \(-0.802204\pi\)
−0.813068 + 0.582169i \(0.802204\pi\)
\(62\) 2.67513 0.339742
\(63\) 1.28726 0.162179
\(64\) 18.1187 2.26484
\(65\) 0 0
\(66\) −7.92478 −0.975473
\(67\) −12.5623 −1.53473 −0.767364 − 0.641211i \(-0.778432\pi\)
−0.767364 + 0.641211i \(0.778432\pi\)
\(68\) 11.1187 1.34834
\(69\) 4.80606 0.578582
\(70\) 0 0
\(71\) −0.481194 −0.0571073 −0.0285536 − 0.999592i \(-0.509090\pi\)
−0.0285536 + 0.999592i \(0.509090\pi\)
\(72\) −8.44358 −0.995086
\(73\) −5.21203 −0.610023 −0.305011 − 0.952349i \(-0.598660\pi\)
−0.305011 + 0.952349i \(0.598660\pi\)
\(74\) 7.05571 0.820210
\(75\) 0 0
\(76\) −12.3127 −1.41236
\(77\) −3.81336 −0.434572
\(78\) 9.83146 1.11319
\(79\) 15.4314 1.73616 0.868082 − 0.496421i \(-0.165353\pi\)
0.868082 + 0.496421i \(0.165353\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −31.1998 −3.44544
\(83\) 10.7308 1.17786 0.588931 − 0.808183i \(-0.299549\pi\)
0.588931 + 0.808183i \(0.299549\pi\)
\(84\) −6.63752 −0.724213
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) −0.168544 −0.0180698
\(88\) 25.0132 2.66641
\(89\) 3.44358 0.365019 0.182510 − 0.983204i \(-0.441578\pi\)
0.182510 + 0.983204i \(0.441578\pi\)
\(90\) 0 0
\(91\) 4.73084 0.495927
\(92\) −24.7816 −2.58366
\(93\) 1.00000 0.103695
\(94\) 33.0191 3.40566
\(95\) 0 0
\(96\) 15.9502 1.62791
\(97\) 15.1998 1.54331 0.771654 − 0.636043i \(-0.219430\pi\)
0.771654 + 0.636043i \(0.219430\pi\)
\(98\) 14.2931 1.44382
\(99\) −2.96239 −0.297731
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.1.1 3
3.2 odd 2 6975.2.a.bi.1.3 3
5.2 odd 4 2325.2.c.l.1024.1 6
5.3 odd 4 2325.2.c.l.1024.6 6
5.4 even 2 465.2.a.g.1.3 ✓ 3
15.14 odd 2 1395.2.a.h.1.1 3
20.19 odd 2 7440.2.a.bm.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.g.1.3 ✓ 3 5.4 even 2
1395.2.a.h.1.1 3 15.14 odd 2
2325.2.a.p.1.1 3 1.1 even 1 trivial
2325.2.c.l.1024.1 6 5.2 odd 4
2325.2.c.l.1024.6 6 5.3 odd 4
6975.2.a.bi.1.3 3 3.2 odd 2
7440.2.a.bm.1.2 3 20.19 odd 2