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 1.1
Root \(-33.9278\) 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)
 
\(f(q)\) \(=\) \(q-30.9278 q^{2} +444.529 q^{4} +625.000 q^{5} +2401.00 q^{7} +2086.72 q^{8} -19329.9 q^{10} +51657.7 q^{11} -129921. q^{13} -74257.7 q^{14} -292137. q^{16} +259513. q^{17} +90316.4 q^{19} +277831. q^{20} -1.59766e6 q^{22} -751692. q^{23} +390625. q^{25} +4.01818e6 q^{26} +1.06731e6 q^{28} +2.20510e6 q^{29} +3.10032e6 q^{31} +7.96675e6 q^{32} -8.02616e6 q^{34} +1.50062e6 q^{35} -1.42524e7 q^{37} -2.79329e6 q^{38} +1.30420e6 q^{40} +1.45691e7 q^{41} +1.89091e7 q^{43} +2.29633e7 q^{44} +2.32482e7 q^{46} +1.43175e7 q^{47} +5.76480e6 q^{49} -1.20812e7 q^{50} -5.77538e7 q^{52} +3.44220e7 q^{53} +3.22860e7 q^{55} +5.01022e6 q^{56} -6.81989e7 q^{58} +3.61330e7 q^{59} -9.33007e7 q^{61} -9.58860e7 q^{62} -9.68200e7 q^{64} -8.12008e7 q^{65} +2.05762e8 q^{67} +1.15361e8 q^{68} -4.64110e7 q^{70} +1.92397e8 q^{71} +8.11669e7 q^{73} +4.40795e8 q^{74} +4.01483e7 q^{76} +1.24030e8 q^{77} +1.07122e8 q^{79} -1.82585e8 q^{80} -4.50592e8 q^{82} -3.54922e8 q^{83} +1.62195e8 q^{85} -5.84818e8 q^{86} +1.07795e8 q^{88} -4.42433e8 q^{89} -3.11941e8 q^{91} -3.34149e8 q^{92} -4.42810e8 q^{94} +5.64478e7 q^{95} -6.40463e8 q^{97} -1.78293e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 13 q^{2} + 501 q^{4} + 2500 q^{5} + 9604 q^{7} + 16263 q^{8} + 8125 q^{10} + 87062 q^{11} + 39494 q^{13} + 31213 q^{14} + 328849 q^{16} + 291756 q^{17} + 50482 q^{19} + 313125 q^{20} - 1003016 q^{22}+ \cdots + 74942413 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −30.9278 −1.36683 −0.683414 − 0.730031i \(-0.739506\pi\)
−0.683414 + 0.730031i \(0.739506\pi\)
\(3\) 0 0
\(4\) 444.529 0.868221
\(5\) 625.000 0.447214
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) 2086.72 0.180119
\(9\) 0 0
\(10\) −19329.9 −0.611264
\(11\) 51657.7 1.06382 0.531909 − 0.846801i \(-0.321475\pi\)
0.531909 + 0.846801i \(0.321475\pi\)
\(12\) 0 0
\(13\) −129921. −1.26164 −0.630820 − 0.775930i \(-0.717281\pi\)
−0.630820 + 0.775930i \(0.717281\pi\)
\(14\) −74257.7 −0.516613
\(15\) 0 0
\(16\) −292137. −1.11441
\(17\) 259513. 0.753595 0.376798 − 0.926296i \(-0.377025\pi\)
0.376798 + 0.926296i \(0.377025\pi\)
\(18\) 0 0
\(19\) 90316.4 0.158992 0.0794961 − 0.996835i \(-0.474669\pi\)
0.0794961 + 0.996835i \(0.474669\pi\)
\(20\) 277831. 0.388280
\(21\) 0 0
\(22\) −1.59766e6 −1.45406
\(23\) −751692. −0.560099 −0.280049 − 0.959986i \(-0.590351\pi\)
−0.280049 + 0.959986i \(0.590351\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 4.01818e6 1.72444
\(27\) 0 0
\(28\) 1.06731e6 0.328157
\(29\) 2.20510e6 0.578945 0.289472 − 0.957186i \(-0.406520\pi\)
0.289472 + 0.957186i \(0.406520\pi\)
\(30\) 0 0
\(31\) 3.10032e6 0.602946 0.301473 − 0.953475i \(-0.402522\pi\)
0.301473 + 0.953475i \(0.402522\pi\)
\(32\) 7.96675e6 1.34309
\(33\) 0 0
\(34\) −8.02616e6 −1.03004
\(35\) 1.50062e6 0.169031
\(36\) 0 0
\(37\) −1.42524e7 −1.25020 −0.625101 − 0.780544i \(-0.714942\pi\)
−0.625101 + 0.780544i \(0.714942\pi\)
\(38\) −2.79329e6 −0.217315
\(39\) 0 0
\(40\) 1.30420e6 0.0805517
\(41\) 1.45691e7 0.805205 0.402603 − 0.915375i \(-0.368106\pi\)
0.402603 + 0.915375i \(0.368106\pi\)
\(42\) 0 0
\(43\) 1.89091e7 0.843458 0.421729 − 0.906722i \(-0.361423\pi\)
0.421729 + 0.906722i \(0.361423\pi\)
\(44\) 2.29633e7 0.923630
\(45\) 0 0
\(46\) 2.32482e7 0.765559
\(47\) 1.43175e7 0.427985 0.213992 − 0.976835i \(-0.431353\pi\)
0.213992 + 0.976835i \(0.431353\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) −1.20812e7 −0.273366
\(51\) 0 0
\(52\) −5.77538e7 −1.09538
\(53\) 3.44220e7 0.599232 0.299616 − 0.954060i \(-0.403142\pi\)
0.299616 + 0.954060i \(0.403142\pi\)
\(54\) 0 0
\(55\) 3.22860e7 0.475754
\(56\) 5.01022e6 0.0680786
\(57\) 0 0
\(58\) −6.81989e7 −0.791319
\(59\) 3.61330e7 0.388213 0.194107 − 0.980980i \(-0.437819\pi\)
0.194107 + 0.980980i \(0.437819\pi\)
\(60\) 0 0
\(61\) −9.33007e7 −0.862781 −0.431390 − 0.902165i \(-0.641977\pi\)
−0.431390 + 0.902165i \(0.641977\pi\)
\(62\) −9.58860e7 −0.824124
\(63\) 0 0
\(64\) −9.68200e7 −0.721365
\(65\) −8.12008e7 −0.564222
\(66\) 0 0
\(67\) 2.05762e8 1.24746 0.623732 − 0.781638i \(-0.285616\pi\)
0.623732 + 0.781638i \(0.285616\pi\)
\(68\) 1.15361e8 0.654288
\(69\) 0 0
\(70\) −4.64110e7 −0.231036
\(71\) 1.92397e8 0.898535 0.449268 − 0.893397i \(-0.351685\pi\)
0.449268 + 0.893397i \(0.351685\pi\)
\(72\) 0 0
\(73\) 8.11669e7 0.334523 0.167262 − 0.985913i \(-0.446508\pi\)
0.167262 + 0.985913i \(0.446508\pi\)
\(74\) 4.40795e8 1.70881
\(75\) 0 0
\(76\) 4.01483e7 0.138040
\(77\) 1.24030e8 0.402086
\(78\) 0 0
\(79\) 1.07122e8 0.309427 0.154713 − 0.987959i \(-0.450555\pi\)
0.154713 + 0.987959i \(0.450555\pi\)
\(80\) −1.82585e8 −0.498381
\(81\) 0 0
\(82\) −4.50592e8 −1.10058
\(83\) −3.54922e8 −0.820883 −0.410441 − 0.911887i \(-0.634625\pi\)
−0.410441 + 0.911887i \(0.634625\pi\)
\(84\) 0 0
\(85\) 1.62195e8 0.337018
\(86\) −5.84818e8 −1.15286
\(87\) 0 0
\(88\) 1.07795e8 0.191614
\(89\) −4.42433e8 −0.747468 −0.373734 − 0.927536i \(-0.621923\pi\)
−0.373734 + 0.927536i \(0.621923\pi\)
\(90\) 0 0
\(91\) −3.11941e8 −0.476855
\(92\) −3.34149e8 −0.486290
\(93\) 0 0
\(94\) −4.42810e8 −0.584982
\(95\) 5.64478e7 0.0711034
\(96\) 0 0
\(97\) −6.40463e8 −0.734549 −0.367275 − 0.930113i \(-0.619709\pi\)
−0.367275 + 0.930113i \(0.619709\pi\)
\(98\) −1.78293e8 −0.195261
\(99\) 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.1.1 4
3.2 odd 2 105.10.a.d.1.4 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.10.a.d.1.4 ✓ 4 3.2 odd 2
315.10.a.e.1.1 4 1.1 even 1 trivial