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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [675,4,Mod(1,675)] 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("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,26,0,0,-30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(39.8262892539\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{58 +6 \sqrt{41}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 29x^{2} + 118 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.21254\) of defining polynomial
Character \(\chi\) \(=\) 675.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.21254 q^{2} -3.10469 q^{4} -17.1047 q^{7} -24.5695 q^{8} +66.8393 q^{11} -72.7328 q^{13} -37.8447 q^{14} -29.5234 q^{16} -40.2889 q^{17} +38.4766 q^{19} +147.884 q^{22} +204.480 q^{23} -160.924 q^{26} +53.1047 q^{28} -21.6621 q^{29} +128.372 q^{31} +131.234 q^{32} -89.1406 q^{34} -19.4187 q^{37} +85.1308 q^{38} +270.393 q^{41} -242.884 q^{43} -207.515 q^{44} +452.419 q^{46} +307.646 q^{47} -50.4297 q^{49} +225.813 q^{52} +289.019 q^{53} +420.254 q^{56} -47.9282 q^{58} +17.7003 q^{59} +764.851 q^{61} +284.027 q^{62} +526.548 q^{64} +532.176 q^{67} +125.084 q^{68} +409.886 q^{71} -220.581 q^{73} -42.9647 q^{74} -119.458 q^{76} -1143.27 q^{77} +1133.70 q^{79} +598.253 q^{82} +253.619 q^{83} -537.390 q^{86} -1642.21 q^{88} -1625.13 q^{89} +1244.07 q^{91} -634.846 q^{92} +680.678 q^{94} -457.573 q^{97} -111.578 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 26 q^{4} - 30 q^{7} - 22 q^{13} + 74 q^{16} + 346 q^{19} - 100 q^{22} + 174 q^{28} + 744 q^{31} + 796 q^{34} + 76 q^{37} - 280 q^{43} + 1656 q^{46} - 778 q^{49} + 2440 q^{52} - 2420 q^{58} + 178 q^{61}+ \cdots + 2050 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.21254 0.782249 0.391125 − 0.920338i \(-0.372086\pi\)
0.391125 + 0.920338i \(0.372086\pi\)
\(3\) 0 0
\(4\) −3.10469 −0.388086
\(5\) 0 0
\(6\) 0 0
\(7\) −17.1047 −0.923566 −0.461783 − 0.886993i \(-0.652790\pi\)
−0.461783 + 0.886993i \(0.652790\pi\)
\(8\) −24.5695 −1.08583
\(9\) 0 0
\(10\) 0 0
\(11\) 66.8393 1.83207 0.916037 − 0.401094i \(-0.131370\pi\)
0.916037 + 0.401094i \(0.131370\pi\)
\(12\) 0 0
\(13\) −72.7328 −1.55173 −0.775863 − 0.630901i \(-0.782685\pi\)
−0.775863 + 0.630901i \(0.782685\pi\)
\(14\) −37.8447 −0.722459
\(15\) 0 0
\(16\) −29.5234 −0.461304
\(17\) −40.2889 −0.574794 −0.287397 − 0.957812i \(-0.592790\pi\)
−0.287397 + 0.957812i \(0.592790\pi\)
\(18\) 0 0
\(19\) 38.4766 0.464586 0.232293 − 0.972646i \(-0.425377\pi\)
0.232293 + 0.972646i \(0.425377\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 147.884 1.43314
\(23\) 204.480 1.85378 0.926891 − 0.375331i \(-0.122471\pi\)
0.926891 + 0.375331i \(0.122471\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −160.924 −1.21384
\(27\) 0 0
\(28\) 53.1047 0.358423
\(29\) −21.6621 −0.138709 −0.0693544 − 0.997592i \(-0.522094\pi\)
−0.0693544 + 0.997592i \(0.522094\pi\)
\(30\) 0 0
\(31\) 128.372 0.743751 0.371875 − 0.928283i \(-0.378715\pi\)
0.371875 + 0.928283i \(0.378715\pi\)
\(32\) 131.234 0.724975
\(33\) 0 0
\(34\) −89.1406 −0.449632
\(35\) 0 0
\(36\) 0 0
\(37\) −19.4187 −0.0862817 −0.0431408 − 0.999069i \(-0.513736\pi\)
−0.0431408 + 0.999069i \(0.513736\pi\)
\(38\) 85.1308 0.363422
\(39\) 0 0
\(40\) 0 0
\(41\) 270.393 1.02996 0.514978 − 0.857203i \(-0.327800\pi\)
0.514978 + 0.857203i \(0.327800\pi\)
\(42\) 0 0
\(43\) −242.884 −0.861384 −0.430692 − 0.902499i \(-0.641730\pi\)
−0.430692 + 0.902499i \(0.641730\pi\)
\(44\) −207.515 −0.711002
\(45\) 0 0
\(46\) 452.419 1.45012
\(47\) 307.646 0.954783 0.477391 − 0.878691i \(-0.341582\pi\)
0.477391 + 0.878691i \(0.341582\pi\)
\(48\) 0 0
\(49\) −50.4297 −0.147025
\(50\) 0 0
\(51\) 0 0
\(52\) 225.813 0.602203
\(53\) 289.019 0.749054 0.374527 − 0.927216i \(-0.377805\pi\)
0.374527 + 0.927216i \(0.377805\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 420.254 1.00284
\(57\) 0 0
\(58\) −47.9282 −0.108505
\(59\) 17.7003 0.0390573 0.0195287 − 0.999809i \(-0.493783\pi\)
0.0195287 + 0.999809i \(0.493783\pi\)
\(60\) 0 0
\(61\) 764.851 1.60540 0.802698 − 0.596385i \(-0.203397\pi\)
0.802698 + 0.596385i \(0.203397\pi\)
\(62\) 284.027 0.581799
\(63\) 0 0
\(64\) 526.548 1.02841
\(65\) 0 0
\(66\) 0 0
\(67\) 532.176 0.970384 0.485192 − 0.874408i \(-0.338750\pi\)
0.485192 + 0.874408i \(0.338750\pi\)
\(68\) 125.084 0.223069
\(69\) 0 0
\(70\) 0 0
\(71\) 409.886 0.685134 0.342567 − 0.939493i \(-0.388704\pi\)
0.342567 + 0.939493i \(0.388704\pi\)
\(72\) 0 0
\(73\) −220.581 −0.353659 −0.176829 − 0.984242i \(-0.556584\pi\)
−0.176829 + 0.984242i \(0.556584\pi\)
\(74\) −42.9647 −0.0674938
\(75\) 0 0
\(76\) −119.458 −0.180299
\(77\) −1143.27 −1.69204
\(78\) 0 0
\(79\) 1133.70 1.61457 0.807286 − 0.590160i \(-0.200935\pi\)
0.807286 + 0.590160i \(0.200935\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 598.253 0.805683
\(83\) 253.619 0.335401 0.167700 − 0.985838i \(-0.446366\pi\)
0.167700 + 0.985838i \(0.446366\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −537.390 −0.673817
\(87\) 0 0
\(88\) −1642.21 −1.98932
\(89\) −1625.13 −1.93555 −0.967775 − 0.251816i \(-0.918972\pi\)
−0.967775 + 0.251816i \(0.918972\pi\)
\(90\) 0 0
\(91\) 1244.07 1.43312
\(92\) −634.846 −0.719426
\(93\) 0 0
\(94\) 680.678 0.746878
\(95\) 0 0
\(96\) 0 0
\(97\) −457.573 −0.478964 −0.239482 − 0.970901i \(-0.576978\pi\)
−0.239482 + 0.970901i \(0.576978\pi\)
\(98\) −111.578 −0.115011
\(99\) 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 675.4.a.w.1.3 yes 4
3.2 odd 2 inner 675.4.a.w.1.2 ✓ 4
5.2 odd 4 675.4.b.o.649.5 8
5.3 odd 4 675.4.b.o.649.4 8
5.4 even 2 675.4.a.x.1.2 yes 4
15.2 even 4 675.4.b.o.649.3 8
15.8 even 4 675.4.b.o.649.6 8
15.14 odd 2 675.4.a.x.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
675.4.a.w.1.2 ✓ 4 3.2 odd 2 inner
675.4.a.w.1.3 yes 4 1.1 even 1 trivial
675.4.a.x.1.2 yes 4 5.4 even 2
675.4.a.x.1.3 yes 4 15.14 odd 2
675.4.b.o.649.3 8 15.2 even 4
675.4.b.o.649.4 8 5.3 odd 4
675.4.b.o.649.5 8 5.2 odd 4
675.4.b.o.649.6 8 15.8 even 4