Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [63,7,Mod(10,63)] 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("63.10"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 63.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,8,0,60,150] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.4934072681\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 10.2
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 63.10
Dual form 63.7.m.b.19.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(4.12132 + 7.13834i) q^{2} +(-1.97056 + 3.41311i) q^{4} +(-68.5660 + 39.5866i) q^{5} +(337.286 + 62.3451i) q^{7} +495.044 q^{8} +(-565.165 - 326.298i) q^{10} +(-854.459 + 1479.97i) q^{11} +3129.09i q^{13} +(945.025 + 2664.61i) q^{14} +(2166.35 + 3752.23i) q^{16} +(-3529.96 - 2038.03i) q^{17} +(5085.89 - 2936.34i) q^{19} -312.032i q^{20} -14086.0 q^{22} +(6660.75 + 11536.8i) q^{23} +(-4678.30 + 8103.05i) q^{25} +(-22336.5 + 12896.0i) q^{26} +(-877.435 + 1028.34i) q^{28} -6510.23 q^{29} +(-10386.6 - 5996.69i) q^{31} +(-2015.04 + 3490.16i) q^{32} -33597.4i q^{34} +(-25594.4 + 9077.27i) q^{35} +(2320.45 + 4019.14i) q^{37} +(41921.2 + 24203.2i) q^{38} +(-33943.2 + 19597.1i) q^{40} -19308.8i q^{41} +91636.4 q^{43} +(-3367.53 - 5832.73i) q^{44} +(-54902.2 + 95093.3i) q^{46} +(55800.2 - 32216.2i) q^{47} +(109875. + 42056.3i) q^{49} -77123.1 q^{50} +(-10680.0 - 6166.07i) q^{52} +(74799.9 - 129557. i) q^{53} -135301. i q^{55} +(166971. + 30863.5i) q^{56} +(-26830.7 - 46472.2i) q^{58} +(-52855.0 - 30515.9i) q^{59} +(-85403.4 + 49307.7i) q^{61} -98857.1i q^{62} +244074. q^{64} +(-123870. - 214549. i) q^{65} +(155906. - 270038. i) q^{67} +(13912.0 - 8032.11i) q^{68} +(-170279. - 145291. i) q^{70} +401209. q^{71} +(-582322. - 336204. i) q^{73} +(-19126.6 + 33128.3i) q^{74} +23145.0i q^{76} +(-380466. + 445901. i) q^{77} +(160076. + 277260. i) q^{79} +(-297076. - 171517. i) q^{80} +(137833. - 79577.6i) q^{82} -832356. i q^{83} +322714. q^{85} +(377663. + 654131. i) q^{86} +(-422995. + 732648. i) q^{88} +(-328654. + 189748. i) q^{89} +(-195084. + 1.05540e6i) q^{91} -52501.7 q^{92} +(459941. + 265547. i) q^{94} +(-232480. + 402666. i) q^{95} -1.05514e6i q^{97} +(152619. + 957653. i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 60 q^{4} + 150 q^{5} + 280 q^{7} + 928 q^{8} - 1200 q^{10} - 1882 q^{11} + 1820 q^{14} + 248 q^{16} - 13458 q^{17} + 18078 q^{19} - 28088 q^{22} - 2470 q^{23} + 2500 q^{25} - 43848 q^{26}+ \cdots + 361424 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.12132 + 7.13834i 0.515165 + 0.892292i 0.999845 + 0.0176005i \(0.00560271\pi\)
−0.484680 + 0.874692i \(0.661064\pi\)
\(3\) 0 0
\(4\) −1.97056 + 3.41311i −0.0307900 + 0.0533299i
\(5\) −68.5660 + 39.5866i −0.548528 + 0.316693i −0.748528 − 0.663103i \(-0.769239\pi\)
0.200000 + 0.979796i \(0.435906\pi\)
\(6\) 0 0
\(7\) 337.286 + 62.3451i 0.983342 + 0.181764i
\(8\) 495.044 0.966882
\(9\) 0 0
\(10\) −565.165 − 326.298i −0.565165 − 0.326298i
\(11\) −854.459 + 1479.97i −0.641968 + 1.11192i 0.343025 + 0.939326i \(0.388548\pi\)
−0.984993 + 0.172594i \(0.944785\pi\)
\(12\) 0 0
\(13\) 3129.09i 1.42426i 0.702049 + 0.712129i \(0.252269\pi\)
−0.702049 + 0.712129i \(0.747731\pi\)
\(14\) 945.025 + 2664.61i 0.344397 + 0.971067i
\(15\) 0 0
\(16\) 2166.35 + 3752.23i 0.528894 + 0.916071i
\(17\) −3529.96 − 2038.03i −0.718494 − 0.414823i 0.0957039 − 0.995410i \(-0.469490\pi\)
−0.814198 + 0.580587i \(0.802823\pi\)
\(18\) 0 0
\(19\) 5085.89 − 2936.34i 0.741492 − 0.428101i −0.0811196 − 0.996704i \(-0.525850\pi\)
0.822611 + 0.568604i \(0.192516\pi\)
\(20\) − 312.032i − 0.0390039i
\(21\) 0 0
\(22\) −14086.0 −1.32288
\(23\) 6660.75 + 11536.8i 0.547444 + 0.948201i 0.998449 + 0.0556791i \(0.0177324\pi\)
−0.451005 + 0.892522i \(0.648934\pi\)
\(24\) 0 0
\(25\) −4678.30 + 8103.05i −0.299411 + 0.518596i
\(26\) −22336.5 + 12896.0i −1.27085 + 0.733728i
\(27\) 0 0
\(28\) −877.435 + 1028.34i −0.0399706 + 0.0468450i
\(29\) −6510.23 −0.266933 −0.133466 − 0.991053i \(-0.542611\pi\)
−0.133466 + 0.991053i \(0.542611\pi\)
\(30\) 0 0
\(31\) −10386.6 − 5996.69i −0.348648 − 0.201292i 0.315442 − 0.948945i \(-0.397847\pi\)
−0.664090 + 0.747653i \(0.731181\pi\)
\(32\) −2015.04 + 3490.16i −0.0614943 + 0.106511i
\(33\) 0 0
\(34\) − 33597.4i − 0.854809i
\(35\) −25594.4 + 9077.27i −0.596954 + 0.211715i
\(36\) 0 0
\(37\) 2320.45 + 4019.14i 0.0458107 + 0.0793465i 0.888022 − 0.459802i \(-0.152080\pi\)
−0.842211 + 0.539148i \(0.818746\pi\)
\(38\) 41921.2 + 24203.2i 0.763981 + 0.441085i
\(39\) 0 0
\(40\) −33943.2 + 19597.1i −0.530362 + 0.306205i
\(41\) − 19308.8i − 0.280158i −0.990140 − 0.140079i \(-0.955264\pi\)
0.990140 − 0.140079i \(-0.0447357\pi\)
\(42\) 0 0
\(43\) 91636.4 1.15256 0.576279 − 0.817253i \(-0.304504\pi\)
0.576279 + 0.817253i \(0.304504\pi\)
\(44\) −3367.53 − 5832.73i −0.0395324 − 0.0684722i
\(45\) 0 0
\(46\) −54902.2 + 95093.3i −0.564048 + 0.976960i
\(47\) 55800.2 − 32216.2i 0.537455 − 0.310300i −0.206592 − 0.978427i \(-0.566237\pi\)
0.744047 + 0.668128i \(0.232904\pi\)
\(48\) 0 0
\(49\) 109875. + 42056.3i 0.933924 + 0.357473i
\(50\) −77123.1 −0.616985
\(51\) 0 0
\(52\) −10680.0 − 6166.07i −0.0759555 − 0.0438529i
\(53\) 74799.9 − 129557.i 0.502428 − 0.870230i −0.497568 − 0.867425i \(-0.665774\pi\)
0.999996 − 0.00280549i \(-0.000893015\pi\)
\(54\) 0 0
\(55\) − 135301.i − 0.813226i
\(56\) 166971. + 30863.5i 0.950776 + 0.175744i
\(57\) 0 0
\(58\) −26830.7 − 46472.2i −0.137515 − 0.238182i
\(59\) −52855.0 − 30515.9i −0.257354 − 0.148583i 0.365773 − 0.930704i \(-0.380805\pi\)
−0.623127 + 0.782121i \(0.714138\pi\)
\(60\) 0 0
\(61\) −85403.4 + 49307.7i −0.376258 + 0.217233i −0.676189 − 0.736728i \(-0.736370\pi\)
0.299931 + 0.953961i \(0.403036\pi\)
\(62\) − 98857.1i − 0.414794i
\(63\) 0 0
\(64\) 244074. 0.931069
\(65\) −123870. − 214549.i −0.451052 − 0.781245i
\(66\) 0 0
\(67\) 155906. − 270038.i 0.518369 − 0.897842i −0.481403 − 0.876499i \(-0.659873\pi\)
0.999772 − 0.0213423i \(-0.00679398\pi\)
\(68\) 13912.0 − 8032.11i 0.0442449 − 0.0255448i
\(69\) 0 0
\(70\) −170279. − 145291.i −0.496441 − 0.423589i
\(71\) 401209. 1.12097 0.560487 − 0.828163i \(-0.310614\pi\)
0.560487 + 0.828163i \(0.310614\pi\)
\(72\) 0 0
\(73\) −582322. − 336204.i −1.49691 − 0.864239i −0.496912 − 0.867801i \(-0.665533\pi\)
−0.999994 + 0.00356186i \(0.998866\pi\)
\(74\) −19126.6 + 33128.3i −0.0472002 + 0.0817531i
\(75\) 0 0
\(76\) 23145.0i 0.0527249i
\(77\) −380466. + 445901.i −0.833381 + 0.976712i
\(78\) 0 0
\(79\) 160076. + 277260.i 0.324672 + 0.562348i 0.981446 − 0.191739i \(-0.0614128\pi\)
−0.656774 + 0.754087i \(0.728079\pi\)
\(80\) −297076. − 171517.i −0.580226 − 0.334994i
\(81\) 0 0
\(82\) 137833. − 79577.6i 0.249983 − 0.144328i
\(83\) − 832356.i − 1.45571i −0.685731 − 0.727855i \(-0.740517\pi\)
0.685731 − 0.727855i \(-0.259483\pi\)
\(84\) 0 0
\(85\) 322714. 0.525486
\(86\) 377663. + 654131.i 0.593757 + 1.02842i
\(87\) 0 0
\(88\) −422995. + 732648.i −0.620707 + 1.07510i
\(89\) −328654. + 189748.i −0.466196 + 0.269158i −0.714646 − 0.699486i \(-0.753412\pi\)
0.248450 + 0.968645i \(0.420079\pi\)
\(90\) 0 0
\(91\) −195084. + 1.05540e6i −0.258879 + 1.40053i
\(92\) −52501.7 −0.0674233
\(93\) 0 0
\(94\) 459941. + 265547.i 0.553756 + 0.319711i
\(95\) −232480. + 402666.i −0.271153 + 0.469650i
\(96\) 0 0
\(97\) − 1.05514e6i − 1.15610i −0.816001 − 0.578050i \(-0.803814\pi\)
0.816001 − 0.578050i \(-0.196186\pi\)
\(98\) 152619. + 957653.i 0.162155 + 1.01749i
\(99\) 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 63.7.m.b.10.2 4
3.2 odd 2 7.7.d.b.3.1 ✓ 4
7.3 odd 6 441.7.d.b.244.2 4
7.4 even 3 441.7.d.b.244.1 4
7.5 odd 6 inner 63.7.m.b.19.2 4
12.11 even 2 112.7.s.b.17.2 4
21.2 odd 6 49.7.d.c.19.1 4
21.5 even 6 7.7.d.b.5.1 yes 4
21.11 odd 6 49.7.b.b.48.4 4
21.17 even 6 49.7.b.b.48.3 4
21.20 even 2 49.7.d.c.31.1 4
84.47 odd 6 112.7.s.b.33.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.7.d.b.3.1 ✓ 4 3.2 odd 2
7.7.d.b.5.1 yes 4 21.5 even 6
49.7.b.b.48.3 4 21.17 even 6
49.7.b.b.48.4 4 21.11 odd 6
49.7.d.c.19.1 4 21.2 odd 6
49.7.d.c.31.1 4 21.20 even 2
63.7.m.b.10.2 4 1.1 even 1 trivial
63.7.m.b.19.2 4 7.5 odd 6 inner
112.7.s.b.17.2 4 12.11 even 2
112.7.s.b.33.2 4 84.47 odd 6
441.7.d.b.244.1 4 7.4 even 3
441.7.d.b.244.2 4 7.3 odd 6