Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,4,0,0,-6,9,0,0,-1,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.4754596827\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2775)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.91223\) of defining polynomial
Character \(\chi\) \(=\) 8325.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.656620 q^{2} -1.56885 q^{4} -1.34338 q^{7} +2.34338 q^{8} -4.16784 q^{11} -3.56885 q^{13} +0.882090 q^{14} +1.59899 q^{16} +0.744391 q^{17} -1.74439 q^{19} +2.73669 q^{22} +4.25561 q^{23} +2.34338 q^{26} +2.10756 q^{28} -1.68676 q^{29} +8.13770 q^{31} -5.73669 q^{32} -0.488783 q^{34} -1.00000 q^{37} +1.14540 q^{38} +11.6791 q^{41} -8.39331 q^{43} +6.53871 q^{44} -2.79432 q^{46} +8.79432 q^{47} -5.19533 q^{49} +5.59899 q^{52} +1.48108 q^{53} -3.14805 q^{56} +1.10756 q^{58} +11.0499 q^{59} -5.59899 q^{61} -5.34338 q^{62} +0.568850 q^{64} +4.28310 q^{67} -1.16784 q^{68} +1.70655 q^{71} +4.87439 q^{73} +0.656620 q^{74} +2.73669 q^{76} +5.59899 q^{77} -7.96986 q^{79} -7.66871 q^{82} +7.56115 q^{83} +5.51122 q^{86} -9.76683 q^{88} -5.37087 q^{89} +4.79432 q^{91} -6.67641 q^{92} -5.77453 q^{94} +1.13770 q^{97} +3.41136 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{4} - 6 q^{7} + 9 q^{8} - q^{11} - 2 q^{13} - 10 q^{14} + 2 q^{16} + 7 q^{17} - 10 q^{19} - 12 q^{22} + 8 q^{23} + 9 q^{26} - 17 q^{28} - 9 q^{29} + 7 q^{31} + 3 q^{32} - 11 q^{34} - 3 q^{37}+ \cdots + 49 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.656620 −0.464301 −0.232150 − 0.972680i \(-0.574576\pi\)
−0.232150 + 0.972680i \(0.574576\pi\)
\(3\) 0 0
\(4\) −1.56885 −0.784425
\(5\) 0 0
\(6\) 0 0
\(7\) −1.34338 −0.507750 −0.253875 − 0.967237i \(-0.581705\pi\)
−0.253875 + 0.967237i \(0.581705\pi\)
\(8\) 2.34338 0.828510
\(9\) 0 0
\(10\) 0 0
\(11\) −4.16784 −1.25665 −0.628325 − 0.777951i \(-0.716259\pi\)
−0.628325 + 0.777951i \(0.716259\pi\)
\(12\) 0 0
\(13\) −3.56885 −0.989821 −0.494910 − 0.868944i \(-0.664799\pi\)
−0.494910 + 0.868944i \(0.664799\pi\)
\(14\) 0.882090 0.235749
\(15\) 0 0
\(16\) 1.59899 0.399747
\(17\) 0.744391 0.180541 0.0902707 − 0.995917i \(-0.471227\pi\)
0.0902707 + 0.995917i \(0.471227\pi\)
\(18\) 0 0
\(19\) −1.74439 −0.400191 −0.200095 − 0.979776i \(-0.564125\pi\)
−0.200095 + 0.979776i \(0.564125\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.73669 0.583464
\(23\) 4.25561 0.887356 0.443678 − 0.896186i \(-0.353673\pi\)
0.443678 + 0.896186i \(0.353673\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.34338 0.459575
\(27\) 0 0
\(28\) 2.10756 0.398291
\(29\) −1.68676 −0.313223 −0.156612 − 0.987660i \(-0.550057\pi\)
−0.156612 + 0.987660i \(0.550057\pi\)
\(30\) 0 0
\(31\) 8.13770 1.46157 0.730787 − 0.682606i \(-0.239153\pi\)
0.730787 + 0.682606i \(0.239153\pi\)
\(32\) −5.73669 −1.01411
\(33\) 0 0
\(34\) −0.488783 −0.0838255
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 1.14540 0.185809
\(39\) 0 0
\(40\) 0 0
\(41\) 11.6791 1.82396 0.911981 − 0.410232i \(-0.134552\pi\)
0.911981 + 0.410232i \(0.134552\pi\)
\(42\) 0 0
\(43\) −8.39331 −1.27997 −0.639984 − 0.768388i \(-0.721059\pi\)
−0.639984 + 0.768388i \(0.721059\pi\)
\(44\) 6.53871 0.985748
\(45\) 0 0
\(46\) −2.79432 −0.412000
\(47\) 8.79432 1.28278 0.641392 − 0.767214i \(-0.278357\pi\)
0.641392 + 0.767214i \(0.278357\pi\)
\(48\) 0 0
\(49\) −5.19533 −0.742190
\(50\) 0 0
\(51\) 0 0
\(52\) 5.59899 0.776440
\(53\) 1.48108 0.203442 0.101721 − 0.994813i \(-0.467565\pi\)
0.101721 + 0.994813i \(0.467565\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.14805 −0.420676
\(57\) 0 0
\(58\) 1.10756 0.145430
\(59\) 11.0499 1.43858 0.719289 − 0.694711i \(-0.244468\pi\)
0.719289 + 0.694711i \(0.244468\pi\)
\(60\) 0 0
\(61\) −5.59899 −0.716877 −0.358438 − 0.933553i \(-0.616691\pi\)
−0.358438 + 0.933553i \(0.616691\pi\)
\(62\) −5.34338 −0.678610
\(63\) 0 0
\(64\) 0.568850 0.0711062
\(65\) 0 0
\(66\) 0 0
\(67\) 4.28310 0.523264 0.261632 − 0.965168i \(-0.415739\pi\)
0.261632 + 0.965168i \(0.415739\pi\)
\(68\) −1.16784 −0.141621
\(69\) 0 0
\(70\) 0 0
\(71\) 1.70655 0.202530 0.101265 − 0.994859i \(-0.467711\pi\)
0.101265 + 0.994859i \(0.467711\pi\)
\(72\) 0 0
\(73\) 4.87439 0.570504 0.285252 − 0.958453i \(-0.407923\pi\)
0.285252 + 0.958453i \(0.407923\pi\)
\(74\) 0.656620 0.0763306
\(75\) 0 0
\(76\) 2.73669 0.313920
\(77\) 5.59899 0.638064
\(78\) 0 0
\(79\) −7.96986 −0.896679 −0.448340 − 0.893863i \(-0.647985\pi\)
−0.448340 + 0.893863i \(0.647985\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −7.66871 −0.846867
\(83\) 7.56115 0.829944 0.414972 − 0.909834i \(-0.363791\pi\)
0.414972 + 0.909834i \(0.363791\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 5.51122 0.594290
\(87\) 0 0
\(88\) −9.76683 −1.04115
\(89\) −5.37087 −0.569311 −0.284656 − 0.958630i \(-0.591879\pi\)
−0.284656 + 0.958630i \(0.591879\pi\)
\(90\) 0 0
\(91\) 4.79432 0.502581
\(92\) −6.67641 −0.696064
\(93\) 0 0
\(94\) −5.77453 −0.595597
\(95\) 0 0
\(96\) 0 0
\(97\) 1.13770 0.115516 0.0577579 − 0.998331i \(-0.481605\pi\)
0.0577579 + 0.998331i \(0.481605\pi\)
\(98\) 3.41136 0.344599
\(99\) 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 8325.2.a.bo.1.2 3
3.2 odd 2 2775.2.a.v.1.2 yes 3
5.4 even 2 8325.2.a.bp.1.2 3
15.14 odd 2 2775.2.a.u.1.2 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2775.2.a.u.1.2 ✓ 3 15.14 odd 2
2775.2.a.v.1.2 yes 3 3.2 odd 2
8325.2.a.bo.1.2 3 1.1 even 1 trivial
8325.2.a.bp.1.2 3 5.4 even 2