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,-6,0,0,7,0,0,6,0,-4,7] 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: 3.3.148.1
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 2325)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48119\) 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)
 
\(f(q)\) \(=\) \(q+1.67513 q^{2} +0.806063 q^{4} +1.19394 q^{7} -2.00000 q^{8} +4.28726 q^{11} +2.09332 q^{13} +2.00000 q^{14} -4.96239 q^{16} -2.83146 q^{17} -0.0376114 q^{19} +7.18172 q^{22} +3.89938 q^{23} +3.50659 q^{26} +0.962389 q^{28} +3.48119 q^{29} -1.00000 q^{31} -4.31265 q^{32} -4.74306 q^{34} +10.4812 q^{37} -0.0630040 q^{38} +4.06300 q^{41} -7.66291 q^{43} +3.45580 q^{44} +6.53198 q^{46} +0.574515 q^{47} -5.57452 q^{49} +1.68735 q^{52} -2.05571 q^{53} -2.38787 q^{56} +5.83146 q^{58} -3.98778 q^{59} -2.70052 q^{61} -1.67513 q^{62} +2.70052 q^{64} +14.8192 q^{67} -2.28233 q^{68} +1.68735 q^{71} +0.649738 q^{73} +17.5574 q^{74} -0.0303172 q^{76} +5.11871 q^{77} +4.86907 q^{79} +6.80606 q^{82} +7.24965 q^{83} -12.8364 q^{86} -8.57452 q^{88} -12.9126 q^{89} +2.49929 q^{91} +3.14315 q^{92} +0.962389 q^{94} +8.24472 q^{97} -9.33804 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{4} + 4 q^{7} - 6 q^{8} + 7 q^{11} + 6 q^{14} - 4 q^{16} + 7 q^{17} - 11 q^{19} - 4 q^{22} + 5 q^{23} - 10 q^{26} - 8 q^{28} + 5 q^{29} - 3 q^{31} + 8 q^{32} - 18 q^{34} + 26 q^{37} + 4 q^{38}+ \cdots + 8 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.67513 1.18450 0.592248 − 0.805756i \(-0.298240\pi\)
0.592248 + 0.805756i \(0.298240\pi\)
\(3\) 0 0
\(4\) 0.806063 0.403032
\(5\) 0 0
\(6\) 0 0
\(7\) 1.19394 0.451266 0.225633 − 0.974212i \(-0.427555\pi\)
0.225633 + 0.974212i \(0.427555\pi\)
\(8\) −2.00000 −0.707107
\(9\) 0 0
\(10\) 0 0
\(11\) 4.28726 1.29266 0.646328 − 0.763059i \(-0.276304\pi\)
0.646328 + 0.763059i \(0.276304\pi\)
\(12\) 0 0
\(13\) 2.09332 0.580583 0.290291 − 0.956938i \(-0.406248\pi\)
0.290291 + 0.956938i \(0.406248\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −4.96239 −1.24060
\(17\) −2.83146 −0.686729 −0.343364 − 0.939202i \(-0.611567\pi\)
−0.343364 + 0.939202i \(0.611567\pi\)
\(18\) 0 0
\(19\) −0.0376114 −0.00862865 −0.00431432 − 0.999991i \(-0.501373\pi\)
−0.00431432 + 0.999991i \(0.501373\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 7.18172 1.53115
\(23\) 3.89938 0.813078 0.406539 − 0.913633i \(-0.366736\pi\)
0.406539 + 0.913633i \(0.366736\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3.50659 0.687698
\(27\) 0 0
\(28\) 0.962389 0.181874
\(29\) 3.48119 0.646442 0.323221 − 0.946324i \(-0.395234\pi\)
0.323221 + 0.946324i \(0.395234\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) −4.31265 −0.762376
\(33\) 0 0
\(34\) −4.74306 −0.813428
\(35\) 0 0
\(36\) 0 0
\(37\) 10.4812 1.72310 0.861549 − 0.507675i \(-0.169495\pi\)
0.861549 + 0.507675i \(0.169495\pi\)
\(38\) −0.0630040 −0.0102206
\(39\) 0 0
\(40\) 0 0
\(41\) 4.06300 0.634535 0.317267 − 0.948336i \(-0.397235\pi\)
0.317267 + 0.948336i \(0.397235\pi\)
\(42\) 0 0
\(43\) −7.66291 −1.16858 −0.584292 − 0.811544i \(-0.698628\pi\)
−0.584292 + 0.811544i \(0.698628\pi\)
\(44\) 3.45580 0.520982
\(45\) 0 0
\(46\) 6.53198 0.963088
\(47\) 0.574515 0.0838017 0.0419008 − 0.999122i \(-0.486659\pi\)
0.0419008 + 0.999122i \(0.486659\pi\)
\(48\) 0 0
\(49\) −5.57452 −0.796359
\(50\) 0 0
\(51\) 0 0
\(52\) 1.68735 0.233993
\(53\) −2.05571 −0.282373 −0.141187 − 0.989983i \(-0.545092\pi\)
−0.141187 + 0.989983i \(0.545092\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.38787 −0.319093
\(57\) 0 0
\(58\) 5.83146 0.765708
\(59\) −3.98778 −0.519165 −0.259582 − 0.965721i \(-0.583585\pi\)
−0.259582 + 0.965721i \(0.583585\pi\)
\(60\) 0 0
\(61\) −2.70052 −0.345767 −0.172883 − 0.984942i \(-0.555308\pi\)
−0.172883 + 0.984942i \(0.555308\pi\)
\(62\) −1.67513 −0.212742
\(63\) 0 0
\(64\) 2.70052 0.337565
\(65\) 0 0
\(66\) 0 0
\(67\) 14.8192 1.81046 0.905229 − 0.424924i \(-0.139699\pi\)
0.905229 + 0.424924i \(0.139699\pi\)
\(68\) −2.28233 −0.276774
\(69\) 0 0
\(70\) 0 0
\(71\) 1.68735 0.200252 0.100126 − 0.994975i \(-0.468075\pi\)
0.100126 + 0.994975i \(0.468075\pi\)
\(72\) 0 0
\(73\) 0.649738 0.0760461 0.0380231 − 0.999277i \(-0.487894\pi\)
0.0380231 + 0.999277i \(0.487894\pi\)
\(74\) 17.5574 2.04100
\(75\) 0 0
\(76\) −0.0303172 −0.00347762
\(77\) 5.11871 0.583332
\(78\) 0 0
\(79\) 4.86907 0.547813 0.273906 − 0.961756i \(-0.411684\pi\)
0.273906 + 0.961756i \(0.411684\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.80606 0.751604
\(83\) 7.24965 0.795752 0.397876 − 0.917439i \(-0.369747\pi\)
0.397876 + 0.917439i \(0.369747\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −12.8364 −1.38418
\(87\) 0 0
\(88\) −8.57452 −0.914046
\(89\) −12.9126 −1.36873 −0.684364 − 0.729140i \(-0.739920\pi\)
−0.684364 + 0.729140i \(0.739920\pi\)
\(90\) 0 0
\(91\) 2.49929 0.261997
\(92\) 3.14315 0.327696
\(93\) 0 0
\(94\) 0.962389 0.0992628
\(95\) 0 0
\(96\) 0 0
\(97\) 8.24472 0.837125 0.418562 − 0.908188i \(-0.362534\pi\)
0.418562 + 0.908188i \(0.362534\pi\)
\(98\) −9.33804 −0.943285
\(99\) 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.bd.1.3 3
3.2 odd 2 2325.2.a.t.1.1 ✓ 3
5.4 even 2 6975.2.a.bc.1.1 3
15.2 even 4 2325.2.c.m.1024.2 6
15.8 even 4 2325.2.c.m.1024.5 6
15.14 odd 2 2325.2.a.u.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2325.2.a.t.1.1 ✓ 3 3.2 odd 2
2325.2.a.u.1.3 yes 3 15.14 odd 2
2325.2.c.m.1024.2 6 15.2 even 4
2325.2.c.m.1024.5 6 15.8 even 4
6975.2.a.bc.1.1 3 5.4 even 2
6975.2.a.bd.1.3 3 1.1 even 1 trivial