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 1.4
Root \(35.8379\) 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)
 
\(f(q)\) \(=\) \(q+38.8379 q^{2} +996.380 q^{4} +625.000 q^{5} +2401.00 q^{7} +18812.3 q^{8} +24273.7 q^{10} -6971.86 q^{11} +45601.2 q^{13} +93249.7 q^{14} +220483. q^{16} +154849. q^{17} -380792. q^{19} +622737. q^{20} -270772. q^{22} +1.66968e6 q^{23} +390625. q^{25} +1.77105e6 q^{26} +2.39231e6 q^{28} +2.23565e6 q^{29} +5.92992e6 q^{31} -1.06882e6 q^{32} +6.01401e6 q^{34} +1.50062e6 q^{35} +4.08925e6 q^{37} -1.47892e7 q^{38} +1.17577e7 q^{40} -1.62091e6 q^{41} +2.61736e7 q^{43} -6.94663e6 q^{44} +6.48468e7 q^{46} +3.12418e7 q^{47} +5.76480e6 q^{49} +1.51710e7 q^{50} +4.54361e7 q^{52} -8.31434e7 q^{53} -4.35742e6 q^{55} +4.51683e7 q^{56} +8.68279e7 q^{58} +3.03472e7 q^{59} -1.10999e8 q^{61} +2.30305e8 q^{62} -1.54398e8 q^{64} +2.85007e7 q^{65} -1.01836e8 q^{67} +1.54289e8 q^{68} +5.82811e7 q^{70} +3.91346e8 q^{71} +2.16261e8 q^{73} +1.58818e8 q^{74} -3.79414e8 q^{76} -1.67394e7 q^{77} +1.00652e8 q^{79} +1.37802e8 q^{80} -6.29525e7 q^{82} +3.07287e8 q^{83} +9.67807e7 q^{85} +1.01653e9 q^{86} -1.31157e8 q^{88} -2.71171e8 q^{89} +1.09488e8 q^{91} +1.66364e9 q^{92} +1.21337e9 q^{94} -2.37995e8 q^{95} -2.51010e8 q^{97} +2.23893e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 13 q^{2} + 501 q^{4} + 2500 q^{5} + 9604 q^{7} + 16263 q^{8} + 8125 q^{10} + 87062 q^{11} + 39494 q^{13} + 31213 q^{14} + 328849 q^{16} + 291756 q^{17} + 50482 q^{19} + 313125 q^{20} - 1003016 q^{22}+ \cdots + 74942413 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 38.8379 1.71641 0.858204 − 0.513309i \(-0.171581\pi\)
0.858204 + 0.513309i \(0.171581\pi\)
\(3\) 0 0
\(4\) 996.380 1.94605
\(5\) 625.000 0.447214
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) 18812.3 1.62382
\(9\) 0 0
\(10\) 24273.7 0.767601
\(11\) −6971.86 −0.143576 −0.0717880 − 0.997420i \(-0.522871\pi\)
−0.0717880 + 0.997420i \(0.522871\pi\)
\(12\) 0 0
\(13\) 45601.2 0.442824 0.221412 − 0.975180i \(-0.428933\pi\)
0.221412 + 0.975180i \(0.428933\pi\)
\(14\) 93249.7 0.648741
\(15\) 0 0
\(16\) 220483. 0.841074
\(17\) 154849. 0.449664 0.224832 − 0.974398i \(-0.427817\pi\)
0.224832 + 0.974398i \(0.427817\pi\)
\(18\) 0 0
\(19\) −380792. −0.670343 −0.335171 − 0.942157i \(-0.608794\pi\)
−0.335171 + 0.942157i \(0.608794\pi\)
\(20\) 622737. 0.870302
\(21\) 0 0
\(22\) −270772. −0.246435
\(23\) 1.66968e6 1.24411 0.622054 − 0.782974i \(-0.286298\pi\)
0.622054 + 0.782974i \(0.286298\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 1.77105e6 0.760066
\(27\) 0 0
\(28\) 2.39231e6 0.735540
\(29\) 2.23565e6 0.586966 0.293483 − 0.955964i \(-0.405186\pi\)
0.293483 + 0.955964i \(0.405186\pi\)
\(30\) 0 0
\(31\) 5.92992e6 1.15324 0.576622 − 0.817011i \(-0.304370\pi\)
0.576622 + 0.817011i \(0.304370\pi\)
\(32\) −1.06882e6 −0.180190
\(33\) 0 0
\(34\) 6.01401e6 0.771807
\(35\) 1.50062e6 0.169031
\(36\) 0 0
\(37\) 4.08925e6 0.358703 0.179352 − 0.983785i \(-0.442600\pi\)
0.179352 + 0.983785i \(0.442600\pi\)
\(38\) −1.47892e7 −1.15058
\(39\) 0 0
\(40\) 1.17577e7 0.726192
\(41\) −1.62091e6 −0.0895840 −0.0447920 − 0.998996i \(-0.514263\pi\)
−0.0447920 + 0.998996i \(0.514263\pi\)
\(42\) 0 0
\(43\) 2.61736e7 1.16750 0.583749 − 0.811934i \(-0.301585\pi\)
0.583749 + 0.811934i \(0.301585\pi\)
\(44\) −6.94663e6 −0.279407
\(45\) 0 0
\(46\) 6.48468e7 2.13540
\(47\) 3.12418e7 0.933891 0.466945 − 0.884286i \(-0.345355\pi\)
0.466945 + 0.884286i \(0.345355\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 1.51710e7 0.343281
\(51\) 0 0
\(52\) 4.54361e7 0.861759
\(53\) −8.31434e7 −1.44739 −0.723696 − 0.690119i \(-0.757558\pi\)
−0.723696 + 0.690119i \(0.757558\pi\)
\(54\) 0 0
\(55\) −4.35742e6 −0.0642091
\(56\) 4.51683e7 0.613744
\(57\) 0 0
\(58\) 8.68279e7 1.00747
\(59\) 3.03472e7 0.326051 0.163025 − 0.986622i \(-0.447875\pi\)
0.163025 + 0.986622i \(0.447875\pi\)
\(60\) 0 0
\(61\) −1.10999e8 −1.02644 −0.513221 − 0.858256i \(-0.671548\pi\)
−0.513221 + 0.858256i \(0.671548\pi\)
\(62\) 2.30305e8 1.97944
\(63\) 0 0
\(64\) −1.54398e8 −1.15035
\(65\) 2.85007e7 0.198037
\(66\) 0 0
\(67\) −1.01836e8 −0.617400 −0.308700 − 0.951160i \(-0.599894\pi\)
−0.308700 + 0.951160i \(0.599894\pi\)
\(68\) 1.54289e8 0.875071
\(69\) 0 0
\(70\) 5.82811e7 0.290126
\(71\) 3.91346e8 1.82767 0.913837 − 0.406082i \(-0.133105\pi\)
0.913837 + 0.406082i \(0.133105\pi\)
\(72\) 0 0
\(73\) 2.16261e8 0.891304 0.445652 − 0.895206i \(-0.352972\pi\)
0.445652 + 0.895206i \(0.352972\pi\)
\(74\) 1.58818e8 0.615681
\(75\) 0 0
\(76\) −3.79414e8 −1.30452
\(77\) −1.67394e7 −0.0542666
\(78\) 0 0
\(79\) 1.00652e8 0.290738 0.145369 − 0.989377i \(-0.453563\pi\)
0.145369 + 0.989377i \(0.453563\pi\)
\(80\) 1.37802e8 0.376140
\(81\) 0 0
\(82\) −6.29525e7 −0.153763
\(83\) 3.07287e8 0.710710 0.355355 − 0.934731i \(-0.384360\pi\)
0.355355 + 0.934731i \(0.384360\pi\)
\(84\) 0 0
\(85\) 9.67807e7 0.201096
\(86\) 1.01653e9 2.00390
\(87\) 0 0
\(88\) −1.31157e8 −0.233141
\(89\) −2.71171e8 −0.458130 −0.229065 − 0.973411i \(-0.573567\pi\)
−0.229065 + 0.973411i \(0.573567\pi\)
\(90\) 0 0
\(91\) 1.09488e8 0.167372
\(92\) 1.66364e9 2.42110
\(93\) 0 0
\(94\) 1.21337e9 1.60294
\(95\) −2.37995e8 −0.299786
\(96\) 0 0
\(97\) −2.51010e8 −0.287885 −0.143942 − 0.989586i \(-0.545978\pi\)
−0.143942 + 0.989586i \(0.545978\pi\)
\(98\) 2.23893e8 0.245201
\(99\) 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.1.4 4
3.2 odd 2 105.10.a.d.1.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.10.a.d.1.1 ✓ 4 3.2 odd 2
315.10.a.e.1.4 4 1.1 even 1 trivial