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,-2,0,-2,0,0,-8,0,0,0,4,0,-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: \(4\)
Coefficient field: \(\Q(\sqrt{14 +2 \sqrt{5}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 555)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.14896\) 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)
 
\(f(q)\) \(=\) \(q-1.61803 q^{2} +0.618034 q^{4} -0.671869 q^{7} +2.23607 q^{8} +3.14896 q^{11} -5.09513 q^{13} +1.08711 q^{14} -4.85410 q^{16} -5.91596 q^{17} -0.179170 q^{19} -5.09513 q^{22} +4.89233 q^{23} +8.24409 q^{26} -0.415238 q^{28} +0.430845 q^{29} -1.70514 q^{31} +3.38197 q^{32} +9.57222 q^{34} -1.00000 q^{37} +0.289903 q^{38} +11.0951 q^{41} +3.00802 q^{43} +1.94617 q^{44} -7.91596 q^{46} +6.48511 q^{47} -6.54859 q^{49} -3.14896 q^{52} +1.64166 q^{53} -1.50234 q^{56} -0.697122 q^{58} +3.40064 q^{59} +4.00306 q^{61} +2.75898 q^{62} +4.23607 q^{64} -8.44688 q^{67} -3.65626 q^{68} +4.41217 q^{71} -5.40064 q^{73} +1.61803 q^{74} -0.110733 q^{76} -2.11569 q^{77} +9.01271 q^{79} -17.9523 q^{82} -5.82389 q^{83} -4.86708 q^{86} +7.04129 q^{88} -2.45653 q^{89} +3.42326 q^{91} +3.02363 q^{92} -10.4931 q^{94} -5.56726 q^{97} +10.5958 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} - 8 q^{7} + 4 q^{11} - 2 q^{13} + 4 q^{14} - 6 q^{16} - 2 q^{17} - 4 q^{19} - 2 q^{22} + 6 q^{26} + 4 q^{28} + 12 q^{29} - 2 q^{31} + 18 q^{32} + 6 q^{34} - 4 q^{37} + 2 q^{38} + 26 q^{41}+ \cdots + 8 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61803 −1.14412 −0.572061 − 0.820211i \(-0.693856\pi\)
−0.572061 + 0.820211i \(0.693856\pi\)
\(3\) 0 0
\(4\) 0.618034 0.309017
\(5\) 0 0
\(6\) 0 0
\(7\) −0.671869 −0.253943 −0.126971 − 0.991906i \(-0.540526\pi\)
−0.126971 + 0.991906i \(0.540526\pi\)
\(8\) 2.23607 0.790569
\(9\) 0 0
\(10\) 0 0
\(11\) 3.14896 0.949448 0.474724 − 0.880135i \(-0.342548\pi\)
0.474724 + 0.880135i \(0.342548\pi\)
\(12\) 0 0
\(13\) −5.09513 −1.41313 −0.706567 − 0.707646i \(-0.749757\pi\)
−0.706567 + 0.707646i \(0.749757\pi\)
\(14\) 1.08711 0.290542
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) −5.91596 −1.43483 −0.717415 − 0.696646i \(-0.754675\pi\)
−0.717415 + 0.696646i \(0.754675\pi\)
\(18\) 0 0
\(19\) −0.179170 −0.0411044 −0.0205522 − 0.999789i \(-0.506542\pi\)
−0.0205522 + 0.999789i \(0.506542\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.09513 −1.08628
\(23\) 4.89233 1.02012 0.510061 − 0.860138i \(-0.329623\pi\)
0.510061 + 0.860138i \(0.329623\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 8.24409 1.61680
\(27\) 0 0
\(28\) −0.415238 −0.0784726
\(29\) 0.430845 0.0800059 0.0400029 − 0.999200i \(-0.487263\pi\)
0.0400029 + 0.999200i \(0.487263\pi\)
\(30\) 0 0
\(31\) −1.70514 −0.306252 −0.153126 − 0.988207i \(-0.548934\pi\)
−0.153126 + 0.988207i \(0.548934\pi\)
\(32\) 3.38197 0.597853
\(33\) 0 0
\(34\) 9.57222 1.64162
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0.289903 0.0470285
\(39\) 0 0
\(40\) 0 0
\(41\) 11.0951 1.73277 0.866384 − 0.499379i \(-0.166438\pi\)
0.866384 + 0.499379i \(0.166438\pi\)
\(42\) 0 0
\(43\) 3.00802 0.458719 0.229359 − 0.973342i \(-0.426337\pi\)
0.229359 + 0.973342i \(0.426337\pi\)
\(44\) 1.94617 0.293395
\(45\) 0 0
\(46\) −7.91596 −1.16714
\(47\) 6.48511 0.945951 0.472975 − 0.881076i \(-0.343180\pi\)
0.472975 + 0.881076i \(0.343180\pi\)
\(48\) 0 0
\(49\) −6.54859 −0.935513
\(50\) 0 0
\(51\) 0 0
\(52\) −3.14896 −0.436682
\(53\) 1.64166 0.225499 0.112750 − 0.993623i \(-0.464034\pi\)
0.112750 + 0.993623i \(0.464034\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.50234 −0.200759
\(57\) 0 0
\(58\) −0.697122 −0.0915365
\(59\) 3.40064 0.442725 0.221363 − 0.975192i \(-0.428950\pi\)
0.221363 + 0.975192i \(0.428950\pi\)
\(60\) 0 0
\(61\) 4.00306 0.512540 0.256270 − 0.966605i \(-0.417506\pi\)
0.256270 + 0.966605i \(0.417506\pi\)
\(62\) 2.75898 0.350390
\(63\) 0 0
\(64\) 4.23607 0.529508
\(65\) 0 0
\(66\) 0 0
\(67\) −8.44688 −1.03195 −0.515976 − 0.856603i \(-0.672570\pi\)
−0.515976 + 0.856603i \(0.672570\pi\)
\(68\) −3.65626 −0.443387
\(69\) 0 0
\(70\) 0 0
\(71\) 4.41217 0.523629 0.261814 − 0.965118i \(-0.415679\pi\)
0.261814 + 0.965118i \(0.415679\pi\)
\(72\) 0 0
\(73\) −5.40064 −0.632097 −0.316048 − 0.948743i \(-0.602356\pi\)
−0.316048 + 0.948743i \(0.602356\pi\)
\(74\) 1.61803 0.188093
\(75\) 0 0
\(76\) −0.110733 −0.0127020
\(77\) −2.11569 −0.241105
\(78\) 0 0
\(79\) 9.01271 1.01401 0.507004 − 0.861943i \(-0.330753\pi\)
0.507004 + 0.861943i \(0.330753\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −17.9523 −1.98250
\(83\) −5.82389 −0.639255 −0.319628 − 0.947543i \(-0.603558\pi\)
−0.319628 + 0.947543i \(0.603558\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.86708 −0.524830
\(87\) 0 0
\(88\) 7.04129 0.750604
\(89\) −2.45653 −0.260392 −0.130196 − 0.991488i \(-0.541561\pi\)
−0.130196 + 0.991488i \(0.541561\pi\)
\(90\) 0 0
\(91\) 3.42326 0.358855
\(92\) 3.02363 0.315235
\(93\) 0 0
\(94\) −10.4931 −1.08228
\(95\) 0 0
\(96\) 0 0
\(97\) −5.56726 −0.565270 −0.282635 − 0.959228i \(-0.591208\pi\)
−0.282635 + 0.959228i \(0.591208\pi\)
\(98\) 10.5958 1.07034
\(99\) 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.bt.1.2 4
3.2 odd 2 2775.2.a.y.1.4 4
5.2 odd 4 1665.2.c.d.334.2 8
5.3 odd 4 1665.2.c.d.334.8 8
5.4 even 2 8325.2.a.bw.1.3 4
15.2 even 4 555.2.c.b.334.7 yes 8
15.8 even 4 555.2.c.b.334.1 ✓ 8
15.14 odd 2 2775.2.a.w.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.c.b.334.1 ✓ 8 15.8 even 4
555.2.c.b.334.7 yes 8 15.2 even 4
1665.2.c.d.334.2 8 5.2 odd 4
1665.2.c.d.334.8 8 5.3 odd 4
2775.2.a.w.1.1 4 15.14 odd 2
2775.2.a.y.1.4 4 3.2 odd 2
8325.2.a.bt.1.2 4 1.1 even 1 trivial
8325.2.a.bw.1.3 4 5.4 even 2