Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [63,10,Mod(1,63)] 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("63.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0] 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: \(32.4472576783\)
Analytic rank: \(1\)
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: \( x^{4} - 479x^{2} + 46396 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(18.5529\) of defining polynomial
Character \(\chi\) \(=\) 63.1

$q$-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)
 
\(f(q)\) \(=\) \(q+37.1058 q^{2} +864.841 q^{4} -1840.64 q^{5} -2401.00 q^{7} +13092.5 q^{8} -68298.4 q^{10} +40288.0 q^{11} -75475.0 q^{13} -89091.1 q^{14} +43007.7 q^{16} -550735. q^{17} -661485. q^{19} -1.59186e6 q^{20} +1.49492e6 q^{22} -1.03047e6 q^{23} +1.43483e6 q^{25} -2.80056e6 q^{26} -2.07648e6 q^{28} +2.37800e6 q^{29} +2.81492e6 q^{31} -5.10751e6 q^{32} -2.04355e7 q^{34} +4.41938e6 q^{35} -1.64714e7 q^{37} -2.45450e7 q^{38} -2.40985e7 q^{40} +2.33528e7 q^{41} +2.96042e7 q^{43} +3.48427e7 q^{44} -3.82365e7 q^{46} +3.95959e6 q^{47} +5.76480e6 q^{49} +5.32405e7 q^{50} -6.52739e7 q^{52} +9.06465e7 q^{53} -7.41557e7 q^{55} -3.14350e7 q^{56} +8.82377e7 q^{58} -6.34335e7 q^{59} +7.23795e7 q^{61} +1.04450e8 q^{62} -2.11538e8 q^{64} +1.38922e8 q^{65} -3.02570e8 q^{67} -4.76298e8 q^{68} +1.63985e8 q^{70} -1.00076e8 q^{71} -2.40030e8 q^{73} -6.11186e8 q^{74} -5.72080e8 q^{76} -9.67315e7 q^{77} +5.38863e8 q^{79} -7.91616e7 q^{80} +8.66525e8 q^{82} -1.03933e8 q^{83} +1.01370e9 q^{85} +1.09849e9 q^{86} +5.27469e8 q^{88} +3.05436e8 q^{89} +1.81216e8 q^{91} -8.91194e8 q^{92} +1.46924e8 q^{94} +1.21756e9 q^{95} -1.10241e9 q^{97} +2.13908e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 1784 q^{4} - 9604 q^{7} - 77176 q^{10} - 291848 q^{13} - 464608 q^{16} + 527200 q^{19} + 660392 q^{22} + 2237804 q^{25} - 4283384 q^{28} - 10657440 q^{31} - 42084264 q^{34} - 26229864 q^{37} - 77006688 q^{40}+ \cdots - 1678668264 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 37.1058 1.63986 0.819930 − 0.572463i \(-0.194012\pi\)
0.819930 + 0.572463i \(0.194012\pi\)
\(3\) 0 0
\(4\) 864.841 1.68914
\(5\) −1840.64 −1.31705 −0.658527 − 0.752557i \(-0.728820\pi\)
−0.658527 + 0.752557i \(0.728820\pi\)
\(6\) 0 0
\(7\) −2401.00 −0.377964
\(8\) 13092.5 1.13010
\(9\) 0 0
\(10\) −68298.4 −2.15979
\(11\) 40288.0 0.829676 0.414838 − 0.909895i \(-0.363838\pi\)
0.414838 + 0.909895i \(0.363838\pi\)
\(12\) 0 0
\(13\) −75475.0 −0.732923 −0.366461 − 0.930433i \(-0.619431\pi\)
−0.366461 + 0.930433i \(0.619431\pi\)
\(14\) −89091.1 −0.619809
\(15\) 0 0
\(16\) 43007.7 0.164061
\(17\) −550735. −1.59927 −0.799636 − 0.600485i \(-0.794974\pi\)
−0.799636 + 0.600485i \(0.794974\pi\)
\(18\) 0 0
\(19\) −661485. −1.16447 −0.582236 − 0.813020i \(-0.697822\pi\)
−0.582236 + 0.813020i \(0.697822\pi\)
\(20\) −1.59186e6 −2.22469
\(21\) 0 0
\(22\) 1.49492e6 1.36055
\(23\) −1.03047e6 −0.767822 −0.383911 − 0.923370i \(-0.625423\pi\)
−0.383911 + 0.923370i \(0.625423\pi\)
\(24\) 0 0
\(25\) 1.43483e6 0.734633
\(26\) −2.80056e6 −1.20189
\(27\) 0 0
\(28\) −2.07648e6 −0.638436
\(29\) 2.37800e6 0.624340 0.312170 − 0.950026i \(-0.398944\pi\)
0.312170 + 0.950026i \(0.398944\pi\)
\(30\) 0 0
\(31\) 2.81492e6 0.547443 0.273721 − 0.961809i \(-0.411745\pi\)
0.273721 + 0.961809i \(0.411745\pi\)
\(32\) −5.10751e6 −0.861061
\(33\) 0 0
\(34\) −2.04355e7 −2.62258
\(35\) 4.41938e6 0.497800
\(36\) 0 0
\(37\) −1.64714e7 −1.44485 −0.722427 − 0.691448i \(-0.756973\pi\)
−0.722427 + 0.691448i \(0.756973\pi\)
\(38\) −2.45450e7 −1.90957
\(39\) 0 0
\(40\) −2.40985e7 −1.48840
\(41\) 2.33528e7 1.29066 0.645330 − 0.763904i \(-0.276720\pi\)
0.645330 + 0.763904i \(0.276720\pi\)
\(42\) 0 0
\(43\) 2.96042e7 1.32052 0.660260 − 0.751038i \(-0.270446\pi\)
0.660260 + 0.751038i \(0.270446\pi\)
\(44\) 3.48427e7 1.40144
\(45\) 0 0
\(46\) −3.82365e7 −1.25912
\(47\) 3.95959e6 0.118361 0.0591807 − 0.998247i \(-0.481151\pi\)
0.0591807 + 0.998247i \(0.481151\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 5.32405e7 1.20470
\(51\) 0 0
\(52\) −6.52739e7 −1.23801
\(53\) 9.06465e7 1.57801 0.789004 − 0.614388i \(-0.210597\pi\)
0.789004 + 0.614388i \(0.210597\pi\)
\(54\) 0 0
\(55\) −7.41557e7 −1.09273
\(56\) −3.14350e7 −0.427137
\(57\) 0 0
\(58\) 8.82377e7 1.02383
\(59\) −6.34335e7 −0.681530 −0.340765 − 0.940149i \(-0.610686\pi\)
−0.340765 + 0.940149i \(0.610686\pi\)
\(60\) 0 0
\(61\) 7.23795e7 0.669317 0.334658 − 0.942340i \(-0.391379\pi\)
0.334658 + 0.942340i \(0.391379\pi\)
\(62\) 1.04450e8 0.897730
\(63\) 0 0
\(64\) −2.11538e8 −1.57608
\(65\) 1.38922e8 0.965299
\(66\) 0 0
\(67\) −3.02570e8 −1.83438 −0.917190 − 0.398451i \(-0.869548\pi\)
−0.917190 + 0.398451i \(0.869548\pi\)
\(68\) −4.76298e8 −2.70140
\(69\) 0 0
\(70\) 1.63985e8 0.816322
\(71\) −1.00076e8 −0.467378 −0.233689 − 0.972311i \(-0.575080\pi\)
−0.233689 + 0.972311i \(0.575080\pi\)
\(72\) 0 0
\(73\) −2.40030e8 −0.989267 −0.494633 − 0.869102i \(-0.664698\pi\)
−0.494633 + 0.869102i \(0.664698\pi\)
\(74\) −6.11186e8 −2.36936
\(75\) 0 0
\(76\) −5.72080e8 −1.96696
\(77\) −9.67315e7 −0.313588
\(78\) 0 0
\(79\) 5.38863e8 1.55653 0.778263 − 0.627939i \(-0.216101\pi\)
0.778263 + 0.627939i \(0.216101\pi\)
\(80\) −7.91616e7 −0.216078
\(81\) 0 0
\(82\) 8.66525e8 2.11650
\(83\) −1.03933e8 −0.240383 −0.120192 − 0.992751i \(-0.538351\pi\)
−0.120192 + 0.992751i \(0.538351\pi\)
\(84\) 0 0
\(85\) 1.01370e9 2.10633
\(86\) 1.09849e9 2.16547
\(87\) 0 0
\(88\) 5.27469e8 0.937616
\(89\) 3.05436e8 0.516019 0.258009 − 0.966142i \(-0.416933\pi\)
0.258009 + 0.966142i \(0.416933\pi\)
\(90\) 0 0
\(91\) 1.81216e8 0.277019
\(92\) −8.91194e8 −1.29696
\(93\) 0 0
\(94\) 1.46924e8 0.194096
\(95\) 1.21756e9 1.53367
\(96\) 0 0
\(97\) −1.10241e9 −1.26436 −0.632179 − 0.774822i \(-0.717839\pi\)
−0.632179 + 0.774822i \(0.717839\pi\)
\(98\) 2.13908e8 0.234266
\(99\) 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 63.10.a.g.1.4 yes 4
3.2 odd 2 inner 63.10.a.g.1.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.10.a.g.1.1 ✓ 4 3.2 odd 2 inner
63.10.a.g.1.4 yes 4 1.1 even 1 trivial