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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,7,Mod(3,49)] 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("49.3"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 49.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.2726500974\)
Analytic rank: \(0\)
Dimension: \(324\)
Relative dimension: \(27\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label 33.10
Character \(\chi\) \(=\) 49.33
Dual form 49.7.h.a.3.10

$q$-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+(-1.80046 - 4.58749i) q^{2} +(34.7561 - 2.60461i) q^{3} +(29.1119 - 27.0119i) q^{4} +(-64.1993 - 94.1630i) q^{5} +(-74.5254 - 154.754i) q^{6} +(-320.804 - 121.382i) q^{7} +(-460.499 - 221.764i) q^{8} +(480.345 - 72.4004i) q^{9} +(-316.384 + 464.049i) q^{10} +(1284.47 + 193.603i) q^{11} +(941.462 - 1014.65i) q^{12} +(-604.180 - 481.818i) q^{13} +(20.7581 + 1690.23i) q^{14} +(-2476.57 - 3105.53i) q^{15} +(1.70409 - 22.7395i) q^{16} +(347.974 + 1128.10i) q^{17} +(-1196.98 - 2073.22i) q^{18} +(1023.34 + 590.825i) q^{19} +(-4412.49 - 1007.12i) q^{20} +(-11466.1 - 3383.18i) q^{21} +(-1424.49 - 6241.08i) q^{22} +(-1745.65 - 538.462i) q^{23} +(-16582.7 - 6508.25i) q^{24} +(963.324 - 2454.51i) q^{25} +(-1122.53 + 3639.16i) q^{26} +(-8264.86 + 1886.40i) q^{27} +(-12618.0 + 5131.90i) q^{28} +(5614.97 - 24600.8i) q^{29} +(-9787.59 + 16952.6i) q^{30} +(40614.4 - 23448.7i) q^{31} +(-31365.5 + 9674.96i) q^{32} +(45147.6 + 3383.34i) q^{33} +(4548.65 - 3627.43i) q^{34} +(9165.76 + 38000.5i) q^{35} +(12028.1 - 15082.8i) q^{36} +(53416.9 + 49563.6i) q^{37} +(867.925 - 5758.30i) q^{38} +(-22253.9 - 15172.4i) q^{39} +(8681.66 + 57599.0i) q^{40} +(17766.2 - 36892.0i) q^{41} +(5123.86 + 58691.7i) q^{42} +(41899.8 - 20177.9i) q^{43} +(42623.1 - 29060.0i) q^{44} +(-37655.2 - 40582.7i) q^{45} +(672.780 + 8977.63i) q^{46} +(-272.088 + 106.787i) q^{47} -794.774i q^{48} +(88182.0 + 77879.5i) q^{49} -12994.5 q^{50} +(15032.5 + 38302.2i) q^{51} +(-30603.7 + 2293.43i) q^{52} +(-133349. + 123730. i) q^{53} +(23534.3 + 34518.5i) q^{54} +(-64232.0 - 133379. i) q^{55} +(120812. + 127039. i) q^{56} +(37106.1 + 17869.4i) q^{57} +(-122965. + 18534.0i) q^{58} +(102339. - 150104. i) q^{59} +(-155984. - 23510.8i) q^{60} +(86928.0 - 93686.1i) q^{61} +(-180695. - 144099. i) q^{62} +(-162885. - 35078.7i) q^{63} +(99945.9 + 125328. i) q^{64} +(-6581.48 + 87823.7i) q^{65} +(-65765.1 - 213205. i) q^{66} +(143871. + 249192. i) q^{67} +(40602.5 + 23441.9i) q^{68} +(-62074.5 - 14168.1i) q^{69} +(157824. - 110466. i) q^{70} +(68055.8 + 298172. i) q^{71} +(-237254. - 73183.2i) q^{72} +(189324. + 74304.2i) q^{73} +(131198. - 334286. i) q^{74} +(27088.4 - 87818.3i) q^{75} +(45750.7 - 10442.3i) q^{76} +(-388565. - 218020. i) q^{77} +(-29536.2 + 129407. i) q^{78} +(437963. - 758573. i) q^{79} +(-2250.62 + 1299.40i) q^{80} +(-620735. + 191471. i) q^{81} +(-201229. - 15080.0i) q^{82} +(-56171.7 + 44795.5i) q^{83} +(-425186. + 211230. i) q^{84} +(83886.1 - 105190. i) q^{85} +(-168005. - 155885. i) q^{86} +(131079. - 869652. i) q^{87} +(-548564. - 374005. i) q^{88} +(86575.3 + 574389. i) q^{89} +(-118376. + 245810. i) q^{90} +(135340. + 227906. i) q^{91} +(-65364.2 + 31477.7i) q^{92} +(1.35052e6 - 920771. i) q^{93} +(979.766 + 1055.94i) q^{94} +(-10063.7 - 134291. i) q^{95} +(-1.06494e6 + 417959. i) q^{96} +1.66917e6i q^{97} +(198503. - 544752. i) q^{98} +631008. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 324 q - 13 q^{2} - 11 q^{3} + 819 q^{4} - 179 q^{5} + 770 q^{6} + 392 q^{7} + 828 q^{8} - 1160 q^{9} - 2594 q^{10} - 5305 q^{11} + 7497 q^{12} - 14 q^{13} - 11403 q^{14} - 6196 q^{15} + 27903 q^{16} - 5107 q^{17}+ \cdots - 4449616 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{41}{42}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.80046 − 4.58749i −0.225057 − 0.573436i 0.773089 − 0.634297i \(-0.218710\pi\)
−0.998146 + 0.0608615i \(0.980615\pi\)
\(3\) 34.7561 − 2.60461i 1.28726 − 0.0964670i 0.586552 − 0.809912i \(-0.300485\pi\)
0.700711 + 0.713445i \(0.252866\pi\)
\(4\) 29.1119 − 27.0119i 0.454874 − 0.422061i
\(5\) −64.1993 − 94.1630i −0.513594 − 0.753304i 0.478597 − 0.878035i \(-0.341146\pi\)
−0.992191 + 0.124731i \(0.960193\pi\)
\(6\) −74.5254 − 154.754i −0.345025 − 0.716452i
\(7\) −320.804 − 121.382i −0.935290 − 0.353882i
\(8\) −460.499 − 221.764i −0.899411 − 0.433134i
\(9\) 480.345 − 72.4004i 0.658910 − 0.0993146i
\(10\) −316.384 + 464.049i −0.316384 + 0.464049i
\(11\) 1284.47 + 193.603i 0.965044 + 0.145457i 0.612615 − 0.790381i \(-0.290118\pi\)
0.352429 + 0.935838i \(0.385356\pi\)
\(12\) 941.462 − 1014.65i 0.544828 − 0.587184i
\(13\) −604.180 − 481.818i −0.275002 − 0.219307i 0.476271 − 0.879299i \(-0.341988\pi\)
−0.751273 + 0.659992i \(0.770560\pi\)
\(14\) 20.7581 + 1690.23i 0.00756491 + 0.615972i
\(15\) −2476.57 − 3105.53i −0.733800 − 0.920156i
\(16\) 1.70409 − 22.7395i 0.000416037 − 0.00555163i
\(17\) 347.974 + 1128.10i 0.0708272 + 0.229616i 0.984064 − 0.177812i \(-0.0569019\pi\)
−0.913237 + 0.407428i \(0.866426\pi\)
\(18\) −1196.98 − 2073.22i −0.205243 − 0.355491i
\(19\) 1023.34 + 590.825i 0.149196 + 0.0861386i 0.572740 − 0.819737i \(-0.305881\pi\)
−0.423543 + 0.905876i \(0.639214\pi\)
\(20\) −4412.49 − 1007.12i −0.551561 − 0.125890i
\(21\) −11466.1 − 3383.18i −1.23810 − 0.365315i
\(22\) −1424.49 − 6241.08i −0.133780 − 0.586127i
\(23\) −1745.65 − 538.462i −0.143474 − 0.0442560i 0.222186 − 0.975004i \(-0.428681\pi\)
−0.365661 + 0.930748i \(0.619157\pi\)
\(24\) −16582.7 − 6508.25i −1.19956 − 0.470793i
\(25\) 963.324 − 2454.51i 0.0616528 − 0.157089i
\(26\) −1122.53 + 3639.16i −0.0638673 + 0.207053i
\(27\) −8264.86 + 1886.40i −0.419898 + 0.0958391i
\(28\) −12618.0 + 5131.90i −0.574799 + 0.233778i
\(29\) 5614.97 − 24600.8i 0.230226 − 1.00868i −0.719228 − 0.694775i \(-0.755504\pi\)
0.949453 − 0.313909i \(-0.101639\pi\)
\(30\) −9787.59 + 16952.6i −0.362503 + 0.627874i
\(31\) 40614.4 − 23448.7i 1.36331 − 0.787107i 0.373247 − 0.927732i \(-0.378244\pi\)
0.990063 + 0.140625i \(0.0449110\pi\)
\(32\) −31365.5 + 9674.96i −0.957198 + 0.295256i
\(33\) 45147.6 + 3383.34i 1.25630 + 0.0941465i
\(34\) 4548.65 − 3627.43i 0.115730 − 0.0922916i
\(35\) 9165.76 + 38000.5i 0.213779 + 0.886310i
\(36\) 12028.1 − 15082.8i 0.257804 − 0.323276i
\(37\) 53416.9 + 49563.6i 1.05457 + 0.978494i 0.999800 − 0.0200055i \(-0.00636838\pi\)
0.0547656 + 0.998499i \(0.482559\pi\)
\(38\) 867.925 − 5758.30i 0.0158173 − 0.104941i
\(39\) −22253.9 − 15172.4i −0.375156 − 0.255777i
\(40\) 8681.66 + 57599.0i 0.135651 + 0.899985i
\(41\) 17766.2 − 36892.0i 0.257777 − 0.535279i −0.731410 − 0.681938i \(-0.761138\pi\)
0.989187 + 0.146658i \(0.0468518\pi\)
\(42\) 5123.86 + 58691.7i 0.0691590 + 0.792189i
\(43\) 41899.8 − 20177.9i 0.526996 − 0.253788i −0.151416 − 0.988470i \(-0.548383\pi\)
0.678411 + 0.734683i \(0.262669\pi\)
\(44\) 42623.1 − 29060.0i 0.500365 − 0.341143i
\(45\) −37655.2 − 40582.7i −0.413226 − 0.445352i
\(46\) 672.780 + 8977.63i 0.00691194 + 0.0922334i
\(47\) −272.088 + 106.787i −0.00262069 + 0.00102855i −0.366651 − 0.930359i \(-0.619496\pi\)
0.364030 + 0.931387i \(0.381400\pi\)
\(48\) − 794.774i − 0.00718654i
\(49\) 88182.0 + 77879.5i 0.749535 + 0.661965i
\(50\) −12994.5 −0.103956
\(51\) 15032.5 + 38302.2i 0.113324 + 0.288744i
\(52\) −30603.7 + 2293.43i −0.217652 + 0.0163108i
\(53\) −133349. + 123730.i −0.895698 + 0.831087i −0.986327 − 0.164798i \(-0.947303\pi\)
0.0906290 + 0.995885i \(0.471112\pi\)
\(54\) 23534.3 + 34518.5i 0.149459 + 0.219215i
\(55\) −64232.0 − 133379.i −0.386068 − 0.801678i
\(56\) 120812. + 127039.i 0.687932 + 0.723391i
\(57\) 37106.1 + 17869.4i 0.200365 + 0.0964905i
\(58\) −122965. + 18534.0i −0.630229 + 0.0949918i
\(59\) 102339. − 150104.i 0.498294 − 0.730863i −0.491887 − 0.870659i \(-0.663693\pi\)
0.990180 + 0.139797i \(0.0446449\pi\)
\(60\) −155984. − 23510.8i −0.722149 − 0.108846i
\(61\) 86928.0 − 93686.1i 0.382975 − 0.412749i −0.511915 − 0.859036i \(-0.671064\pi\)
0.894890 + 0.446288i \(0.147254\pi\)
\(62\) −180695. − 144099.i −0.758178 − 0.604627i
\(63\) −162885. − 35078.7i −0.651417 − 0.140288i
\(64\) 99945.9 + 125328.i 0.381264 + 0.478089i
\(65\) −6581.48 + 87823.7i −0.0239653 + 0.319795i
\(66\) −65765.1 − 213205.i −0.228752 − 0.741594i
\(67\) 143871. + 249192.i 0.478354 + 0.828534i 0.999692 − 0.0248165i \(-0.00790014\pi\)
−0.521338 + 0.853350i \(0.674567\pi\)
\(68\) 40602.5 + 23441.9i 0.129130 + 0.0745530i
\(69\) −62074.5 − 14168.1i −0.188958 − 0.0431285i
\(70\) 157824. − 110466.i 0.460129 − 0.322058i
\(71\) 68055.8 + 298172.i 0.190147 + 0.833089i 0.976535 + 0.215357i \(0.0690914\pi\)
−0.786388 + 0.617732i \(0.788051\pi\)
\(72\) −237254. − 73183.2i −0.635647 − 0.196071i
\(73\) 189324. + 74304.2i 0.486673 + 0.191005i 0.595974 − 0.803004i \(-0.296766\pi\)
−0.109301 + 0.994009i \(0.534861\pi\)
\(74\) 131198. − 334286.i 0.323766 − 0.824942i
\(75\) 27088.4 − 87818.3i 0.0642094 − 0.208162i
\(76\) 45750.7 − 10442.3i 0.104221 − 0.0237879i
\(77\) −388565. − 218020.i −0.851122 − 0.477556i
\(78\) −29536.2 + 129407.i −0.0622402 + 0.272692i
\(79\) 437963. − 758573.i 0.888292 − 1.53857i 0.0463986 − 0.998923i \(-0.485226\pi\)
0.841893 − 0.539644i \(-0.181441\pi\)
\(80\) −2250.62 + 1299.40i −0.00439574 + 0.00253788i
\(81\) −620735. + 191471.i −1.16802 + 0.360287i
\(82\) −201229. − 15080.0i −0.364963 − 0.0273502i
\(83\) −56171.7 + 44795.5i −0.0982389 + 0.0783429i −0.671378 − 0.741115i \(-0.734297\pi\)
0.573139 + 0.819458i \(0.305726\pi\)
\(84\) −425186. + 211230.i −0.717366 + 0.356383i
\(85\) 83886.1 − 105190.i 0.136594 − 0.171284i
\(86\) −168005. − 155885.i −0.264135 − 0.245081i
\(87\) 131079. − 869652.i 0.199056 − 1.32065i
\(88\) −548564. − 374005.i −0.804969 − 0.548819i
\(89\) 86575.3 + 574389.i 0.122807 + 0.814773i 0.962416 + 0.271581i \(0.0875464\pi\)
−0.839608 + 0.543192i \(0.817216\pi\)
\(90\) −118376. + 245810.i −0.162381 + 0.337188i
\(91\) 135340. + 227906.i 0.179598 + 0.302434i
\(92\) −65364.2 + 31477.7i −0.0839415 + 0.0404241i
\(93\) 1.35052e6 − 920771.i 1.67901 − 1.14473i
\(94\) 979.766 + 1055.94i 0.00117961 + 0.00127132i
\(95\) −10063.7 − 134291.i −0.0117378 − 0.156631i
\(96\) −1.06494e6 + 417959.i −1.20368 + 0.472411i
\(97\) 1.66917e6i 1.82888i 0.404721 + 0.914440i \(0.367369\pi\)
−0.404721 + 0.914440i \(0.632631\pi\)
\(98\) 198503. − 544752.i 0.210906 − 0.578790i
\(99\) 631008. 0.650323
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 49.7.h.a.33.10 yes 324
49.3 odd 42 inner 49.7.h.a.3.10 ✓ 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.10 ✓ 324 49.3 odd 42 inner
49.7.h.a.33.10 yes 324 1.1 even 1 trivial