Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [315,10,Mod(1,315)] 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("315.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(162.236288392\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 1253x^{2} - 1039x + 42996 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(5.50890\) of defining polynomial
Character \(\chi\) \(=\) 315.1

$q$-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)
 
\(f(q)\) \(=\) \(q+8.50890 q^{2} -439.599 q^{4} +625.000 q^{5} +2401.00 q^{7} -8097.06 q^{8} +5318.06 q^{10} +84700.6 q^{11} +189194. q^{13} +20429.9 q^{14} +156177. q^{16} +238796. q^{17} +621510. q^{19} -274749. q^{20} +720709. q^{22} +211156. q^{23} +390625. q^{25} +1.60983e6 q^{26} -1.05548e6 q^{28} -3.28769e6 q^{29} -7.79081e6 q^{31} +5.47459e6 q^{32} +2.03189e6 q^{34} +1.50062e6 q^{35} -8.19742e6 q^{37} +5.28836e6 q^{38} -5.06066e6 q^{40} -1.82676e7 q^{41} +8.54852e6 q^{43} -3.72343e7 q^{44} +1.79671e6 q^{46} +2.44728e7 q^{47} +5.76480e6 q^{49} +3.32379e6 q^{50} -8.31693e7 q^{52} +3.12652e7 q^{53} +5.29379e7 q^{55} -1.94410e7 q^{56} -2.79746e7 q^{58} +1.53039e8 q^{59} +9.34546e7 q^{61} -6.62913e7 q^{62} -3.33800e7 q^{64} +1.18246e8 q^{65} -2.45728e6 q^{67} -1.04974e8 q^{68} +1.27687e7 q^{70} +2.39479e8 q^{71} -3.22135e8 q^{73} -6.97510e7 q^{74} -2.73215e8 q^{76} +2.03366e8 q^{77} -6.41950e8 q^{79} +9.76108e7 q^{80} -1.55437e8 q^{82} +7.07851e8 q^{83} +1.49247e8 q^{85} +7.27385e7 q^{86} -6.85826e8 q^{88} -3.67866e8 q^{89} +4.54254e8 q^{91} -9.28240e7 q^{92} +2.08236e8 q^{94} +3.88443e8 q^{95} -5.71196e8 q^{97} +4.90521e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 13 q^{2} + 501 q^{4} + 2500 q^{5} + 9604 q^{7} + 16263 q^{8} + 8125 q^{10} + 87062 q^{11} + 39494 q^{13} + 31213 q^{14} + 328849 q^{16} + 291756 q^{17} + 50482 q^{19} + 313125 q^{20} - 1003016 q^{22}+ \cdots + 74942413 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.50890 0.376044 0.188022 − 0.982165i \(-0.439792\pi\)
0.188022 + 0.982165i \(0.439792\pi\)
\(3\) 0 0
\(4\) −439.599 −0.858591
\(5\) 625.000 0.447214
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) −8097.06 −0.698912
\(9\) 0 0
\(10\) 5318.06 0.168172
\(11\) 84700.6 1.74429 0.872146 − 0.489245i \(-0.162728\pi\)
0.872146 + 0.489245i \(0.162728\pi\)
\(12\) 0 0
\(13\) 189194. 1.83722 0.918611 − 0.395163i \(-0.129312\pi\)
0.918611 + 0.395163i \(0.129312\pi\)
\(14\) 20429.9 0.142131
\(15\) 0 0
\(16\) 156177. 0.595769
\(17\) 238796. 0.693435 0.346718 − 0.937970i \(-0.387296\pi\)
0.346718 + 0.937970i \(0.387296\pi\)
\(18\) 0 0
\(19\) 621510. 1.09410 0.547049 − 0.837100i \(-0.315751\pi\)
0.547049 + 0.837100i \(0.315751\pi\)
\(20\) −274749. −0.383974
\(21\) 0 0
\(22\) 720709. 0.655931
\(23\) 211156. 0.157336 0.0786681 − 0.996901i \(-0.474933\pi\)
0.0786681 + 0.996901i \(0.474933\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 1.60983e6 0.690876
\(27\) 0 0
\(28\) −1.05548e6 −0.324517
\(29\) −3.28769e6 −0.863176 −0.431588 − 0.902071i \(-0.642047\pi\)
−0.431588 + 0.902071i \(0.642047\pi\)
\(30\) 0 0
\(31\) −7.79081e6 −1.51515 −0.757574 − 0.652749i \(-0.773616\pi\)
−0.757574 + 0.652749i \(0.773616\pi\)
\(32\) 5.47459e6 0.922947
\(33\) 0 0
\(34\) 2.03189e6 0.260762
\(35\) 1.50062e6 0.169031
\(36\) 0 0
\(37\) −8.19742e6 −0.719067 −0.359533 − 0.933132i \(-0.617064\pi\)
−0.359533 + 0.933132i \(0.617064\pi\)
\(38\) 5.28836e6 0.411429
\(39\) 0 0
\(40\) −5.06066e6 −0.312563
\(41\) −1.82676e7 −1.00961 −0.504805 − 0.863233i \(-0.668436\pi\)
−0.504805 + 0.863233i \(0.668436\pi\)
\(42\) 0 0
\(43\) 8.54852e6 0.381314 0.190657 − 0.981657i \(-0.438938\pi\)
0.190657 + 0.981657i \(0.438938\pi\)
\(44\) −3.72343e7 −1.49763
\(45\) 0 0
\(46\) 1.79671e6 0.0591653
\(47\) 2.44728e7 0.731548 0.365774 − 0.930704i \(-0.380804\pi\)
0.365774 + 0.930704i \(0.380804\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 3.32379e6 0.0752088
\(51\) 0 0
\(52\) −8.31693e7 −1.57742
\(53\) 3.12652e7 0.544276 0.272138 − 0.962258i \(-0.412269\pi\)
0.272138 + 0.962258i \(0.412269\pi\)
\(54\) 0 0
\(55\) 5.29379e7 0.780071
\(56\) −1.94410e7 −0.264164
\(57\) 0 0
\(58\) −2.79746e7 −0.324592
\(59\) 1.53039e8 1.64426 0.822128 − 0.569303i \(-0.192787\pi\)
0.822128 + 0.569303i \(0.192787\pi\)
\(60\) 0 0
\(61\) 9.34546e7 0.864204 0.432102 − 0.901825i \(-0.357772\pi\)
0.432102 + 0.901825i \(0.357772\pi\)
\(62\) −6.62913e7 −0.569762
\(63\) 0 0
\(64\) −3.33800e7 −0.248701
\(65\) 1.18246e8 0.821631
\(66\) 0 0
\(67\) −2.45728e6 −0.0148976 −0.00744881 − 0.999972i \(-0.502371\pi\)
−0.00744881 + 0.999972i \(0.502371\pi\)
\(68\) −1.04974e8 −0.595377
\(69\) 0 0
\(70\) 1.27687e7 0.0635630
\(71\) 2.39479e8 1.11842 0.559211 − 0.829026i \(-0.311104\pi\)
0.559211 + 0.829026i \(0.311104\pi\)
\(72\) 0 0
\(73\) −3.22135e8 −1.32765 −0.663827 − 0.747886i \(-0.731069\pi\)
−0.663827 + 0.747886i \(0.731069\pi\)
\(74\) −6.97510e7 −0.270401
\(75\) 0 0
\(76\) −2.73215e8 −0.939383
\(77\) 2.03366e8 0.659281
\(78\) 0 0
\(79\) −6.41950e8 −1.85430 −0.927149 − 0.374693i \(-0.877748\pi\)
−0.927149 + 0.374693i \(0.877748\pi\)
\(80\) 9.76108e7 0.266436
\(81\) 0 0
\(82\) −1.55437e8 −0.379658
\(83\) 7.07851e8 1.63716 0.818579 − 0.574394i \(-0.194762\pi\)
0.818579 + 0.574394i \(0.194762\pi\)
\(84\) 0 0
\(85\) 1.49247e8 0.310114
\(86\) 7.27385e7 0.143391
\(87\) 0 0
\(88\) −6.85826e8 −1.21911
\(89\) −3.67866e8 −0.621490 −0.310745 − 0.950493i \(-0.600579\pi\)
−0.310745 + 0.950493i \(0.600579\pi\)
\(90\) 0 0
\(91\) 4.54254e8 0.694405
\(92\) −9.28240e7 −0.135087
\(93\) 0 0
\(94\) 2.08236e8 0.275094
\(95\) 3.88443e8 0.489296
\(96\) 0 0
\(97\) −5.71196e8 −0.655107 −0.327553 − 0.944833i \(-0.606224\pi\)
−0.327553 + 0.944833i \(0.606224\pi\)
\(98\) 4.90521e7 0.0537206
\(99\) 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 315.10.a.e.1.3 4
3.2 odd 2 105.10.a.d.1.2 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.10.a.d.1.2 ✓ 4 3.2 odd 2
315.10.a.e.1.3 4 1.1 even 1 trivial