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 1.1
Root \(2.11491\) 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)
 
\(f(q)\) \(=\) \(q-2.11491 q^{2} +2.47283 q^{4} -1.64207 q^{7} -1.00000 q^{8} -2.94567 q^{11} -4.94567 q^{13} +3.47283 q^{14} -2.83076 q^{16} -1.28415 q^{17} -6.81756 q^{19} +6.22982 q^{22} +2.22982 q^{23} +10.4596 q^{26} -4.06058 q^{28} +5.66152 q^{29} -1.00000 q^{31} +7.98680 q^{32} +2.71585 q^{34} +4.22982 q^{37} +14.4185 q^{38} -4.81756 q^{41} -9.51396 q^{43} -7.28415 q^{44} -4.71585 q^{46} -4.45963 q^{47} -4.30359 q^{49} -12.2298 q^{52} -11.1755 q^{53} +1.64207 q^{56} -11.9736 q^{58} -12.5877 q^{59} -9.74378 q^{61} +2.11491 q^{62} -11.2298 q^{64} -4.00000 q^{67} -3.17548 q^{68} +16.3315 q^{71} -8.68945 q^{73} -8.94567 q^{74} -16.8587 q^{76} +4.83700 q^{77} +3.28415 q^{79} +10.1887 q^{82} +7.28415 q^{83} +20.1212 q^{86} +2.94567 q^{88} +6.00000 q^{89} +8.12115 q^{91} +5.51396 q^{92} +9.43171 q^{94} -5.87189 q^{97} +9.10170 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{4} - 4 q^{7} - 3 q^{8} + 2 q^{11} - 4 q^{13} + 5 q^{14} - 4 q^{16} - 2 q^{17} + 4 q^{19} + 6 q^{22} - 6 q^{23} + 6 q^{26} + 5 q^{28} + 8 q^{29} - 3 q^{31} + 4 q^{32} + 10 q^{34} + 27 q^{38} + 10 q^{41}+ \cdots + q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.11491 −1.49547 −0.747733 − 0.664000i \(-0.768858\pi\)
−0.747733 + 0.664000i \(0.768858\pi\)
\(3\) 0 0
\(4\) 2.47283 1.23642
\(5\) 0 0
\(6\) 0 0
\(7\) −1.64207 −0.620645 −0.310323 − 0.950631i \(-0.600437\pi\)
−0.310323 + 0.950631i \(0.600437\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −2.94567 −0.888152 −0.444076 − 0.895989i \(-0.646468\pi\)
−0.444076 + 0.895989i \(0.646468\pi\)
\(12\) 0 0
\(13\) −4.94567 −1.37168 −0.685841 − 0.727752i \(-0.740565\pi\)
−0.685841 + 0.727752i \(0.740565\pi\)
\(14\) 3.47283 0.928154
\(15\) 0 0
\(16\) −2.83076 −0.707690
\(17\) −1.28415 −0.311451 −0.155726 − 0.987800i \(-0.549772\pi\)
−0.155726 + 0.987800i \(0.549772\pi\)
\(18\) 0 0
\(19\) −6.81756 −1.56405 −0.782027 − 0.623244i \(-0.785814\pi\)
−0.782027 + 0.623244i \(0.785814\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.22982 1.32820
\(23\) 2.22982 0.464949 0.232474 − 0.972603i \(-0.425318\pi\)
0.232474 + 0.972603i \(0.425318\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 10.4596 2.05130
\(27\) 0 0
\(28\) −4.06058 −0.767377
\(29\) 5.66152 1.05132 0.525659 − 0.850695i \(-0.323819\pi\)
0.525659 + 0.850695i \(0.323819\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 7.98680 1.41188
\(33\) 0 0
\(34\) 2.71585 0.465765
\(35\) 0 0
\(36\) 0 0
\(37\) 4.22982 0.695377 0.347689 − 0.937610i \(-0.386967\pi\)
0.347689 + 0.937610i \(0.386967\pi\)
\(38\) 14.4185 2.33899
\(39\) 0 0
\(40\) 0 0
\(41\) −4.81756 −0.752376 −0.376188 − 0.926543i \(-0.622765\pi\)
−0.376188 + 0.926543i \(0.622765\pi\)
\(42\) 0 0
\(43\) −9.51396 −1.45087 −0.725433 − 0.688293i \(-0.758360\pi\)
−0.725433 + 0.688293i \(0.758360\pi\)
\(44\) −7.28415 −1.09813
\(45\) 0 0
\(46\) −4.71585 −0.695315
\(47\) −4.45963 −0.650504 −0.325252 − 0.945627i \(-0.605449\pi\)
−0.325252 + 0.945627i \(0.605449\pi\)
\(48\) 0 0
\(49\) −4.30359 −0.614799
\(50\) 0 0
\(51\) 0 0
\(52\) −12.2298 −1.69597
\(53\) −11.1755 −1.53507 −0.767535 − 0.641007i \(-0.778517\pi\)
−0.767535 + 0.641007i \(0.778517\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.64207 0.219431
\(57\) 0 0
\(58\) −11.9736 −1.57221
\(59\) −12.5877 −1.63878 −0.819392 − 0.573233i \(-0.805689\pi\)
−0.819392 + 0.573233i \(0.805689\pi\)
\(60\) 0 0
\(61\) −9.74378 −1.24756 −0.623781 − 0.781599i \(-0.714404\pi\)
−0.623781 + 0.781599i \(0.714404\pi\)
\(62\) 2.11491 0.268594
\(63\) 0 0
\(64\) −11.2298 −1.40373
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −3.17548 −0.385084
\(69\) 0 0
\(70\) 0 0
\(71\) 16.3315 1.93819 0.969097 − 0.246679i \(-0.0793392\pi\)
0.969097 + 0.246679i \(0.0793392\pi\)
\(72\) 0 0
\(73\) −8.68945 −1.01702 −0.508511 − 0.861055i \(-0.669804\pi\)
−0.508511 + 0.861055i \(0.669804\pi\)
\(74\) −8.94567 −1.03991
\(75\) 0 0
\(76\) −16.8587 −1.93382
\(77\) 4.83700 0.551228
\(78\) 0 0
\(79\) 3.28415 0.369495 0.184748 − 0.982786i \(-0.440853\pi\)
0.184748 + 0.982786i \(0.440853\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 10.1887 1.12515
\(83\) 7.28415 0.799539 0.399770 − 0.916616i \(-0.369090\pi\)
0.399770 + 0.916616i \(0.369090\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 20.1212 2.16972
\(87\) 0 0
\(88\) 2.94567 0.314009
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 8.12115 0.851328
\(92\) 5.51396 0.574870
\(93\) 0 0
\(94\) 9.43171 0.972806
\(95\) 0 0
\(96\) 0 0
\(97\) −5.87189 −0.596200 −0.298100 − 0.954535i \(-0.596353\pi\)
−0.298100 + 0.954535i \(0.596353\pi\)
\(98\) 9.10170 0.919411
\(99\) 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.1.1 3
3.2 odd 2 2325.2.a.s.1.3 3
5.4 even 2 279.2.a.c.1.3 3
15.2 even 4 2325.2.c.n.1024.6 6
15.8 even 4 2325.2.c.n.1024.1 6
15.14 odd 2 93.2.a.b.1.1 ✓ 3
20.19 odd 2 4464.2.a.bq.1.2 3
60.59 even 2 1488.2.a.t.1.2 3
105.104 even 2 4557.2.a.v.1.1 3
120.29 odd 2 5952.2.a.bz.1.2 3
120.59 even 2 5952.2.a.cf.1.2 3
155.154 odd 2 8649.2.a.p.1.3 3
465.464 even 2 2883.2.a.f.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
93.2.a.b.1.1 ✓ 3 15.14 odd 2
279.2.a.c.1.3 3 5.4 even 2
1488.2.a.t.1.2 3 60.59 even 2
2325.2.a.s.1.3 3 3.2 odd 2
2325.2.c.n.1024.1 6 15.8 even 4
2325.2.c.n.1024.6 6 15.2 even 4
2883.2.a.f.1.1 3 465.464 even 2
4464.2.a.bq.1.2 3 20.19 odd 2
4557.2.a.v.1.1 3 105.104 even 2
5952.2.a.bz.1.2 3 120.29 odd 2
5952.2.a.cf.1.2 3 120.59 even 2
6975.2.a.bb.1.1 3 1.1 even 1 trivial
8649.2.a.p.1.3 3 155.154 odd 2