Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6975,2,Mod(1,6975)] 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("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6975.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,2,0,0,-4,-3,0,0,2,0,-4,5,0,-4,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(55.6956554098\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 93)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 6975.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.86081 q^{2} +1.46260 q^{4} +1.32340 q^{7} -1.00000 q^{8} -0.925197 q^{11} -2.92520 q^{13} +2.46260 q^{14} -4.78600 q^{16} +4.64681 q^{17} +6.11982 q^{19} -1.72161 q^{22} -5.72161 q^{23} -5.44322 q^{26} +1.93561 q^{28} +9.57201 q^{29} -1.00000 q^{31} -6.90582 q^{32} +8.64681 q^{34} -3.72161 q^{37} +11.3878 q^{38} +8.11982 q^{41} +4.36842 q^{43} -1.35319 q^{44} -10.6468 q^{46} +11.4432 q^{47} -5.24860 q^{49} -4.27839 q^{52} -1.20359 q^{53} -1.32340 q^{56} +17.8116 q^{58} -7.60179 q^{59} +12.0900 q^{61} -1.86081 q^{62} -3.27839 q^{64} -4.00000 q^{67} +6.79641 q^{68} -10.4882 q^{71} +15.1648 q^{73} -6.92520 q^{74} +8.95084 q^{76} -1.22441 q^{77} -2.64681 q^{79} +15.1094 q^{82} +1.35319 q^{83} +8.12878 q^{86} +0.925197 q^{88} +6.00000 q^{89} -3.87122 q^{91} -8.36842 q^{92} +21.2936 q^{94} +5.04502 q^{97} -9.76663 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{4} - 4 q^{7} - 3 q^{8} + 2 q^{11} - 4 q^{13} + 5 q^{14} - 4 q^{16} - 2 q^{17} + 4 q^{19} + 6 q^{22} - 6 q^{23} + 6 q^{26} + 5 q^{28} + 8 q^{29} - 3 q^{31} + 4 q^{32} + 10 q^{34} + 27 q^{38} + 10 q^{41}+ \cdots + q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.86081 1.31579 0.657894 − 0.753110i \(-0.271447\pi\)
0.657894 + 0.753110i \(0.271447\pi\)
\(3\) 0 0
\(4\) 1.46260 0.731299
\(5\) 0 0
\(6\) 0 0
\(7\) 1.32340 0.500200 0.250100 − 0.968220i \(-0.419537\pi\)
0.250100 + 0.968220i \(0.419537\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −0.925197 −0.278957 −0.139479 − 0.990225i \(-0.544543\pi\)
−0.139479 + 0.990225i \(0.544543\pi\)
\(12\) 0 0
\(13\) −2.92520 −0.811304 −0.405652 − 0.914028i \(-0.632955\pi\)
−0.405652 + 0.914028i \(0.632955\pi\)
\(14\) 2.46260 0.658157
\(15\) 0 0
\(16\) −4.78600 −1.19650
\(17\) 4.64681 1.12702 0.563508 − 0.826110i \(-0.309451\pi\)
0.563508 + 0.826110i \(0.309451\pi\)
\(18\) 0 0
\(19\) 6.11982 1.40398 0.701991 − 0.712185i \(-0.252294\pi\)
0.701991 + 0.712185i \(0.252294\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.72161 −0.367049
\(23\) −5.72161 −1.19304 −0.596519 − 0.802599i \(-0.703450\pi\)
−0.596519 + 0.802599i \(0.703450\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −5.44322 −1.06750
\(27\) 0 0
\(28\) 1.93561 0.365796
\(29\) 9.57201 1.77748 0.888738 − 0.458415i \(-0.151583\pi\)
0.888738 + 0.458415i \(0.151583\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) −6.90582 −1.22079
\(33\) 0 0
\(34\) 8.64681 1.48292
\(35\) 0 0
\(36\) 0 0
\(37\) −3.72161 −0.611829 −0.305915 − 0.952059i \(-0.598962\pi\)
−0.305915 + 0.952059i \(0.598962\pi\)
\(38\) 11.3878 1.84734
\(39\) 0 0
\(40\) 0 0
\(41\) 8.11982 1.26810 0.634051 − 0.773291i \(-0.281391\pi\)
0.634051 + 0.773291i \(0.281391\pi\)
\(42\) 0 0
\(43\) 4.36842 0.666178 0.333089 − 0.942895i \(-0.391909\pi\)
0.333089 + 0.942895i \(0.391909\pi\)
\(44\) −1.35319 −0.204001
\(45\) 0 0
\(46\) −10.6468 −1.56979
\(47\) 11.4432 1.66917 0.834583 − 0.550882i \(-0.185709\pi\)
0.834583 + 0.550882i \(0.185709\pi\)
\(48\) 0 0
\(49\) −5.24860 −0.749800
\(50\) 0 0
\(51\) 0 0
\(52\) −4.27839 −0.593306
\(53\) −1.20359 −0.165325 −0.0826626 − 0.996578i \(-0.526342\pi\)
−0.0826626 + 0.996578i \(0.526342\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.32340 −0.176847
\(57\) 0 0
\(58\) 17.8116 2.33878
\(59\) −7.60179 −0.989669 −0.494835 − 0.868987i \(-0.664771\pi\)
−0.494835 + 0.868987i \(0.664771\pi\)
\(60\) 0 0
\(61\) 12.0900 1.54797 0.773985 − 0.633204i \(-0.218261\pi\)
0.773985 + 0.633204i \(0.218261\pi\)
\(62\) −1.86081 −0.236323
\(63\) 0 0
\(64\) −3.27839 −0.409799
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.79641 0.824186
\(69\) 0 0
\(70\) 0 0
\(71\) −10.4882 −1.24473 −0.622363 − 0.782729i \(-0.713827\pi\)
−0.622363 + 0.782729i \(0.713827\pi\)
\(72\) 0 0
\(73\) 15.1648 1.77491 0.887455 − 0.460895i \(-0.152471\pi\)
0.887455 + 0.460895i \(0.152471\pi\)
\(74\) −6.92520 −0.805038
\(75\) 0 0
\(76\) 8.95084 1.02673
\(77\) −1.22441 −0.139534
\(78\) 0 0
\(79\) −2.64681 −0.297789 −0.148895 − 0.988853i \(-0.547572\pi\)
−0.148895 + 0.988853i \(0.547572\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 15.1094 1.66855
\(83\) 1.35319 0.148532 0.0742660 − 0.997238i \(-0.476339\pi\)
0.0742660 + 0.997238i \(0.476339\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.12878 0.876549
\(87\) 0 0
\(88\) 0.925197 0.0986263
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −3.87122 −0.405814
\(92\) −8.36842 −0.872468
\(93\) 0 0
\(94\) 21.2936 2.19627
\(95\) 0 0
\(96\) 0 0
\(97\) 5.04502 0.512244 0.256122 − 0.966644i \(-0.417555\pi\)
0.256122 + 0.966644i \(0.417555\pi\)
\(98\) −9.76663 −0.986578
\(99\) 0 0
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 6975.2.a.bb.1.3 3
3.2 odd 2 2325.2.a.s.1.1 3
5.4 even 2 279.2.a.c.1.1 3
15.2 even 4 2325.2.c.n.1024.2 6
15.8 even 4 2325.2.c.n.1024.5 6
15.14 odd 2 93.2.a.b.1.3 ✓ 3
20.19 odd 2 4464.2.a.bq.1.3 3
60.59 even 2 1488.2.a.t.1.1 3
105.104 even 2 4557.2.a.v.1.3 3
120.29 odd 2 5952.2.a.bz.1.3 3
120.59 even 2 5952.2.a.cf.1.3 3
155.154 odd 2 8649.2.a.p.1.1 3
465.464 even 2 2883.2.a.f.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
93.2.a.b.1.3 ✓ 3 15.14 odd 2
279.2.a.c.1.1 3 5.4 even 2
1488.2.a.t.1.1 3 60.59 even 2
2325.2.a.s.1.1 3 3.2 odd 2
2325.2.c.n.1024.2 6 15.2 even 4
2325.2.c.n.1024.5 6 15.8 even 4
2883.2.a.f.1.3 3 465.464 even 2
4464.2.a.bq.1.3 3 20.19 odd 2
4557.2.a.v.1.3 3 105.104 even 2
5952.2.a.bz.1.3 3 120.29 odd 2
5952.2.a.cf.1.3 3 120.59 even 2
6975.2.a.bb.1.3 3 1.1 even 1 trivial
8649.2.a.p.1.1 3 155.154 odd 2