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 = [5,-2,0,4,0,0,0,-6,0,0,12,0,3] 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: \(5\)
Coefficient field: 5.5.457904.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 8x^{2} + 5x - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1665)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.788997\) 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)
 
\(f(q)\) \(=\) \(q-0.788997 q^{2} -1.37748 q^{4} -4.52826 q^{7} +2.66482 q^{8} +3.48264 q^{11} +5.61840 q^{13} +3.57278 q^{14} +0.652431 q^{16} -3.78568 q^{17} -1.21100 q^{19} -2.74779 q^{22} -4.09014 q^{23} -4.43290 q^{26} +6.23760 q^{28} -4.02139 q^{29} -2.48074 q^{31} -5.84441 q^{32} +2.98689 q^{34} -1.00000 q^{37} +0.955478 q^{38} +12.6195 q^{41} -0.120859 q^{43} -4.79728 q^{44} +3.22711 q^{46} -4.57799 q^{47} +13.5051 q^{49} -7.73926 q^{52} -4.83920 q^{53} -12.0670 q^{56} +3.17286 q^{58} +0.709346 q^{59} +8.77566 q^{61} +1.95729 q^{62} +3.30636 q^{64} -8.01089 q^{67} +5.21472 q^{68} +2.98579 q^{71} +2.05732 q^{73} +0.788997 q^{74} +1.66814 q^{76} -15.7703 q^{77} -1.46045 q^{79} -9.95675 q^{82} +6.74036 q^{83} +0.0953571 q^{86} +9.28061 q^{88} -1.40670 q^{89} -25.4416 q^{91} +5.63411 q^{92} +3.61202 q^{94} +18.7525 q^{97} -10.6555 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 4 q^{4} - 6 q^{8} + 12 q^{11} + 3 q^{13} + 10 q^{14} - 2 q^{16} - 4 q^{17} - 8 q^{19} - 10 q^{22} - 18 q^{23} + 2 q^{26} + 4 q^{28} + 13 q^{29} - 16 q^{31} - 14 q^{32} - 2 q^{34} - 5 q^{37}+ \cdots + 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.788997 −0.557905 −0.278952 − 0.960305i \(-0.589987\pi\)
−0.278952 + 0.960305i \(0.589987\pi\)
\(3\) 0 0
\(4\) −1.37748 −0.688742
\(5\) 0 0
\(6\) 0 0
\(7\) −4.52826 −1.71152 −0.855760 − 0.517372i \(-0.826910\pi\)
−0.855760 + 0.517372i \(0.826910\pi\)
\(8\) 2.66482 0.942158
\(9\) 0 0
\(10\) 0 0
\(11\) 3.48264 1.05005 0.525027 − 0.851085i \(-0.324055\pi\)
0.525027 + 0.851085i \(0.324055\pi\)
\(12\) 0 0
\(13\) 5.61840 1.55826 0.779132 − 0.626859i \(-0.215660\pi\)
0.779132 + 0.626859i \(0.215660\pi\)
\(14\) 3.57278 0.954866
\(15\) 0 0
\(16\) 0.652431 0.163108
\(17\) −3.78568 −0.918163 −0.459081 − 0.888394i \(-0.651821\pi\)
−0.459081 + 0.888394i \(0.651821\pi\)
\(18\) 0 0
\(19\) −1.21100 −0.277823 −0.138912 − 0.990305i \(-0.544360\pi\)
−0.138912 + 0.990305i \(0.544360\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.74779 −0.585830
\(23\) −4.09014 −0.852854 −0.426427 − 0.904522i \(-0.640228\pi\)
−0.426427 + 0.904522i \(0.640228\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.43290 −0.869363
\(27\) 0 0
\(28\) 6.23760 1.17880
\(29\) −4.02139 −0.746753 −0.373377 − 0.927680i \(-0.621800\pi\)
−0.373377 + 0.927680i \(0.621800\pi\)
\(30\) 0 0
\(31\) −2.48074 −0.445554 −0.222777 − 0.974869i \(-0.571512\pi\)
−0.222777 + 0.974869i \(0.571512\pi\)
\(32\) −5.84441 −1.03316
\(33\) 0 0
\(34\) 2.98689 0.512248
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0.955478 0.154999
\(39\) 0 0
\(40\) 0 0
\(41\) 12.6195 1.97084 0.985418 − 0.170153i \(-0.0544262\pi\)
0.985418 + 0.170153i \(0.0544262\pi\)
\(42\) 0 0
\(43\) −0.120859 −0.0184308 −0.00921539 − 0.999958i \(-0.502933\pi\)
−0.00921539 + 0.999958i \(0.502933\pi\)
\(44\) −4.79728 −0.723217
\(45\) 0 0
\(46\) 3.22711 0.475811
\(47\) −4.57799 −0.667769 −0.333884 − 0.942614i \(-0.608360\pi\)
−0.333884 + 0.942614i \(0.608360\pi\)
\(48\) 0 0
\(49\) 13.5051 1.92930
\(50\) 0 0
\(51\) 0 0
\(52\) −7.73926 −1.07324
\(53\) −4.83920 −0.664715 −0.332358 − 0.943153i \(-0.607844\pi\)
−0.332358 + 0.943153i \(0.607844\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −12.0670 −1.61252
\(57\) 0 0
\(58\) 3.17286 0.416617
\(59\) 0.709346 0.0923490 0.0461745 − 0.998933i \(-0.485297\pi\)
0.0461745 + 0.998933i \(0.485297\pi\)
\(60\) 0 0
\(61\) 8.77566 1.12361 0.561804 − 0.827270i \(-0.310108\pi\)
0.561804 + 0.827270i \(0.310108\pi\)
\(62\) 1.95729 0.248577
\(63\) 0 0
\(64\) 3.30636 0.413295
\(65\) 0 0
\(66\) 0 0
\(67\) −8.01089 −0.978687 −0.489343 − 0.872091i \(-0.662763\pi\)
−0.489343 + 0.872091i \(0.662763\pi\)
\(68\) 5.21472 0.632377
\(69\) 0 0
\(70\) 0 0
\(71\) 2.98579 0.354348 0.177174 − 0.984180i \(-0.443304\pi\)
0.177174 + 0.984180i \(0.443304\pi\)
\(72\) 0 0
\(73\) 2.05732 0.240791 0.120395 − 0.992726i \(-0.461584\pi\)
0.120395 + 0.992726i \(0.461584\pi\)
\(74\) 0.788997 0.0917190
\(75\) 0 0
\(76\) 1.66814 0.191349
\(77\) −15.7703 −1.79719
\(78\) 0 0
\(79\) −1.46045 −0.164313 −0.0821567 − 0.996619i \(-0.526181\pi\)
−0.0821567 + 0.996619i \(0.526181\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −9.95675 −1.09954
\(83\) 6.74036 0.739851 0.369925 − 0.929061i \(-0.379383\pi\)
0.369925 + 0.929061i \(0.379383\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.0953571 0.0102826
\(87\) 0 0
\(88\) 9.28061 0.989317
\(89\) −1.40670 −0.149110 −0.0745550 − 0.997217i \(-0.523754\pi\)
−0.0745550 + 0.997217i \(0.523754\pi\)
\(90\) 0 0
\(91\) −25.4416 −2.66700
\(92\) 5.63411 0.587397
\(93\) 0 0
\(94\) 3.61202 0.372552
\(95\) 0 0
\(96\) 0 0
\(97\) 18.7525 1.90403 0.952015 − 0.306051i \(-0.0990077\pi\)
0.952015 + 0.306051i \(0.0990077\pi\)
\(98\) −10.6555 −1.07637
\(99\) 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.by.1.3 5
3.2 odd 2 8325.2.a.cf.1.3 5
5.4 even 2 1665.2.a.r.1.3 yes 5
15.14 odd 2 1665.2.a.o.1.3 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1665.2.a.o.1.3 ✓ 5 15.14 odd 2
1665.2.a.r.1.3 yes 5 5.4 even 2
8325.2.a.by.1.3 5 1.1 even 1 trivial
8325.2.a.cf.1.3 5 3.2 odd 2