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 = [4,0,0,4,0,0,-4,-6,0,0,0,0,-4] 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: \(4\)
Coefficient field: 4.4.6224.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - 2x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 111)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.796815\) 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)
 
\(f(q)\) \(=\) \(q-0.796815 q^{2} -1.36509 q^{4} -4.63253 q^{7} +2.68135 q^{8} +4.73017 q^{11} -3.69127 q^{13} +3.69127 q^{14} +0.593630 q^{16} -3.88454 q^{17} -4.32380 q^{19} -3.76907 q^{22} +4.51707 q^{23} +2.94126 q^{26} +6.32380 q^{28} +7.07180 q^{29} -2.63253 q^{31} -5.83572 q^{32} +3.09526 q^{34} +1.00000 q^{37} +3.44527 q^{38} -8.22616 q^{41} +5.81979 q^{43} -6.45709 q^{44} -3.59927 q^{46} -0.961099 q^{47} +14.4603 q^{49} +5.03890 q^{52} +9.91743 q^{53} -12.4214 q^{56} -5.63491 q^{58} +5.28289 q^{59} +1.18726 q^{61} +2.09764 q^{62} +3.46273 q^{64} +4.63253 q^{67} +5.30272 q^{68} +10.4214 q^{71} -11.3627 q^{73} -0.796815 q^{74} +5.90236 q^{76} -21.9127 q^{77} -5.13654 q^{79} +6.55473 q^{82} -2.30873 q^{83} -4.63730 q^{86} +12.6833 q^{88} -12.4194 q^{89} +17.0999 q^{91} -6.16618 q^{92} +0.765818 q^{94} +15.0341 q^{97} -11.5222 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 4 q^{7} - 6 q^{8} - 4 q^{13} + 4 q^{14} - 4 q^{16} - 2 q^{17} + 8 q^{19} + 12 q^{22} - 10 q^{23} + 8 q^{26} + 2 q^{29} + 4 q^{31} - 12 q^{32} - 16 q^{34} + 4 q^{37} + 12 q^{38} - 12 q^{41}+ \cdots + 24 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.796815 −0.563433 −0.281717 − 0.959498i \(-0.590904\pi\)
−0.281717 + 0.959498i \(0.590904\pi\)
\(3\) 0 0
\(4\) −1.36509 −0.682543
\(5\) 0 0
\(6\) 0 0
\(7\) −4.63253 −1.75093 −0.875466 − 0.483280i \(-0.839445\pi\)
−0.875466 + 0.483280i \(0.839445\pi\)
\(8\) 2.68135 0.948001
\(9\) 0 0
\(10\) 0 0
\(11\) 4.73017 1.42620 0.713100 − 0.701062i \(-0.247290\pi\)
0.713100 + 0.701062i \(0.247290\pi\)
\(12\) 0 0
\(13\) −3.69127 −1.02377 −0.511887 − 0.859053i \(-0.671053\pi\)
−0.511887 + 0.859053i \(0.671053\pi\)
\(14\) 3.69127 0.986534
\(15\) 0 0
\(16\) 0.593630 0.148408
\(17\) −3.88454 −0.942138 −0.471069 − 0.882096i \(-0.656132\pi\)
−0.471069 + 0.882096i \(0.656132\pi\)
\(18\) 0 0
\(19\) −4.32380 −0.991948 −0.495974 − 0.868337i \(-0.665189\pi\)
−0.495974 + 0.868337i \(0.665189\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.76907 −0.803569
\(23\) 4.51707 0.941874 0.470937 − 0.882167i \(-0.343916\pi\)
0.470937 + 0.882167i \(0.343916\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.94126 0.576829
\(27\) 0 0
\(28\) 6.32380 1.19509
\(29\) 7.07180 1.31320 0.656600 − 0.754239i \(-0.271994\pi\)
0.656600 + 0.754239i \(0.271994\pi\)
\(30\) 0 0
\(31\) −2.63253 −0.472817 −0.236408 − 0.971654i \(-0.575970\pi\)
−0.236408 + 0.971654i \(0.575970\pi\)
\(32\) −5.83572 −1.03162
\(33\) 0 0
\(34\) 3.09526 0.530832
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 3.44527 0.558897
\(39\) 0 0
\(40\) 0 0
\(41\) −8.22616 −1.28471 −0.642355 − 0.766407i \(-0.722043\pi\)
−0.642355 + 0.766407i \(0.722043\pi\)
\(42\) 0 0
\(43\) 5.81979 0.887510 0.443755 − 0.896148i \(-0.353646\pi\)
0.443755 + 0.896148i \(0.353646\pi\)
\(44\) −6.45709 −0.973443
\(45\) 0 0
\(46\) −3.59927 −0.530683
\(47\) −0.961099 −0.140191 −0.0700954 − 0.997540i \(-0.522330\pi\)
−0.0700954 + 0.997540i \(0.522330\pi\)
\(48\) 0 0
\(49\) 14.4603 2.06576
\(50\) 0 0
\(51\) 0 0
\(52\) 5.03890 0.698770
\(53\) 9.91743 1.36226 0.681132 − 0.732161i \(-0.261488\pi\)
0.681132 + 0.732161i \(0.261488\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −12.4214 −1.65989
\(57\) 0 0
\(58\) −5.63491 −0.739900
\(59\) 5.28289 0.687773 0.343887 − 0.939011i \(-0.388256\pi\)
0.343887 + 0.939011i \(0.388256\pi\)
\(60\) 0 0
\(61\) 1.18726 0.152013 0.0760066 − 0.997107i \(-0.475783\pi\)
0.0760066 + 0.997107i \(0.475783\pi\)
\(62\) 2.09764 0.266401
\(63\) 0 0
\(64\) 3.46273 0.432841
\(65\) 0 0
\(66\) 0 0
\(67\) 4.63253 0.565954 0.282977 − 0.959127i \(-0.408678\pi\)
0.282977 + 0.959127i \(0.408678\pi\)
\(68\) 5.30272 0.643050
\(69\) 0 0
\(70\) 0 0
\(71\) 10.4214 1.23680 0.618399 − 0.785864i \(-0.287782\pi\)
0.618399 + 0.785864i \(0.287782\pi\)
\(72\) 0 0
\(73\) −11.3627 −1.32990 −0.664952 − 0.746886i \(-0.731548\pi\)
−0.664952 + 0.746886i \(0.731548\pi\)
\(74\) −0.796815 −0.0926279
\(75\) 0 0
\(76\) 5.90236 0.677047
\(77\) −21.9127 −2.49718
\(78\) 0 0
\(79\) −5.13654 −0.577906 −0.288953 − 0.957343i \(-0.593307\pi\)
−0.288953 + 0.957343i \(0.593307\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.55473 0.723849
\(83\) −2.30873 −0.253416 −0.126708 − 0.991940i \(-0.540441\pi\)
−0.126708 + 0.991940i \(0.540441\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.63730 −0.500053
\(87\) 0 0
\(88\) 12.6833 1.35204
\(89\) −12.4194 −1.31646 −0.658228 − 0.752818i \(-0.728694\pi\)
−0.658228 + 0.752818i \(0.728694\pi\)
\(90\) 0 0
\(91\) 17.0999 1.79256
\(92\) −6.16618 −0.642869
\(93\) 0 0
\(94\) 0.765818 0.0789881
\(95\) 0 0
\(96\) 0 0
\(97\) 15.0341 1.52649 0.763243 − 0.646112i \(-0.223606\pi\)
0.763243 + 0.646112i \(0.223606\pi\)
\(98\) −11.5222 −1.16392
\(99\) 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.bv.1.2 4
3.2 odd 2 2775.2.a.x.1.3 4
5.4 even 2 333.2.a.g.1.3 4
15.14 odd 2 111.2.a.b.1.2 ✓ 4
20.19 odd 2 5328.2.a.bs.1.2 4
60.59 even 2 1776.2.a.u.1.3 4
105.104 even 2 5439.2.a.u.1.2 4
120.29 odd 2 7104.2.a.cc.1.2 4
120.59 even 2 7104.2.a.cf.1.2 4
555.554 odd 2 4107.2.a.i.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
111.2.a.b.1.2 ✓ 4 15.14 odd 2
333.2.a.g.1.3 4 5.4 even 2
1776.2.a.u.1.3 4 60.59 even 2
2775.2.a.x.1.3 4 3.2 odd 2
4107.2.a.i.1.3 4 555.554 odd 2
5328.2.a.bs.1.2 4 20.19 odd 2
5439.2.a.u.1.2 4 105.104 even 2
7104.2.a.cc.1.2 4 120.29 odd 2
7104.2.a.cf.1.2 4 120.59 even 2
8325.2.a.bv.1.2 4 1.1 even 1 trivial