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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,7,Mod(48,49)] 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("49.48"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 49.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.2726500974\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 7 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 48.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 49.48
Dual form 49.7.b.a.48.2

$q$-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)
 
\(f(q)\) \(=\) \(q-12.0000 q^{2} -12.1244i q^{3} +80.0000 q^{4} -181.865i q^{5} +145.492i q^{6} -192.000 q^{8} +582.000 q^{9} +2182.38i q^{10} +1479.00 q^{11} -969.948i q^{12} -484.974i q^{13} -2205.00 q^{15} -2816.00 q^{16} -3018.96i q^{17} -6984.00 q^{18} -6874.51i q^{19} -14549.2i q^{20} -17748.0 q^{22} -5913.00 q^{23} +2327.88i q^{24} -17450.0 q^{25} +5819.69i q^{26} -15895.0i q^{27} +3978.00 q^{29} +26460.0 q^{30} -12815.4i q^{31} +46080.0 q^{32} -17931.9i q^{33} +36227.6i q^{34} +46560.0 q^{36} -61577.0 q^{37} +82494.1i q^{38} -5880.00 q^{39} +34918.1i q^{40} +110574. i q^{41} -17414.0 q^{43} +118320. q^{44} -105846. i q^{45} +70956.0 q^{46} +30662.5i q^{47} +34142.2i q^{48} +209400. q^{50} -36603.0 q^{51} -38797.9i q^{52} -60513.0 q^{53} +190740. i q^{54} -268979. i q^{55} -83349.0 q^{57} -47736.0 q^{58} -215729. i q^{59} -176400. q^{60} -162745. i q^{61} +153785. i q^{62} -372736. q^{64} -88200.0 q^{65} +215183. i q^{66} -268777. q^{67} -241517. i q^{68} +71691.3i q^{69} +101922. q^{71} -111744. q^{72} +317646. i q^{73} +738924. q^{74} +211570. i q^{75} -549961. i q^{76} +70560.0 q^{78} +362231. q^{79} +512133. i q^{80} +231561. q^{81} -1.32689e6i q^{82} -216783. i q^{83} -549045. q^{85} +208968. q^{86} -48230.7i q^{87} -283968. q^{88} +1.33456e6i q^{89} +1.27015e6i q^{90} -473040. q^{92} -155379. q^{93} -367950. i q^{94} -1.25024e6 q^{95} -558690. i q^{96} -1.51409e6i q^{97} +860778. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 24 q^{2} + 160 q^{4} - 384 q^{8} + 1164 q^{9} + 2958 q^{11} - 4410 q^{15} - 5632 q^{16} - 13968 q^{18} - 35496 q^{22} - 11826 q^{23} - 34900 q^{25} + 7956 q^{29} + 52920 q^{30} + 92160 q^{32} + 93120 q^{36}+ \cdots + 1721556 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −12.0000 −1.50000 −0.750000 − 0.661438i \(-0.769947\pi\)
−0.750000 + 0.661438i \(0.769947\pi\)
\(3\) − 12.1244i − 0.449050i −0.974468 − 0.224525i \(-0.927917\pi\)
0.974468 − 0.224525i \(-0.0720831\pi\)
\(4\) 80.0000 1.25000
\(5\) − 181.865i − 1.45492i −0.686149 − 0.727461i \(-0.740700\pi\)
0.686149 − 0.727461i \(-0.259300\pi\)
\(6\) 145.492i 0.673575i
\(7\) 0 0
\(8\) −192.000 −0.375000
\(9\) 582.000 0.798354
\(10\) 2182.38i 2.18238i
\(11\) 1479.00 1.11119 0.555597 − 0.831452i \(-0.312490\pi\)
0.555597 + 0.831452i \(0.312490\pi\)
\(12\) − 969.948i − 0.561313i
\(13\) − 484.974i − 0.220744i −0.993890 − 0.110372i \(-0.964796\pi\)
0.993890 − 0.110372i \(-0.0352042\pi\)
\(14\) 0 0
\(15\) −2205.00 −0.653333
\(16\) −2816.00 −0.687500
\(17\) − 3018.96i − 0.614485i −0.951631 − 0.307242i \(-0.900594\pi\)
0.951631 − 0.307242i \(-0.0994063\pi\)
\(18\) −6984.00 −1.19753
\(19\) − 6874.51i − 1.00226i −0.865372 − 0.501131i \(-0.832918\pi\)
0.865372 − 0.501131i \(-0.167082\pi\)
\(20\) − 14549.2i − 1.81865i
\(21\) 0 0
\(22\) −17748.0 −1.66679
\(23\) −5913.00 −0.485987 −0.242993 − 0.970028i \(-0.578129\pi\)
−0.242993 + 0.970028i \(0.578129\pi\)
\(24\) 2327.88i 0.168394i
\(25\) −17450.0 −1.11680
\(26\) 5819.69i 0.331116i
\(27\) − 15895.0i − 0.807551i
\(28\) 0 0
\(29\) 3978.00 0.163106 0.0815532 − 0.996669i \(-0.474012\pi\)
0.0815532 + 0.996669i \(0.474012\pi\)
\(30\) 26460.0 0.980000
\(31\) − 12815.4i − 0.430178i −0.976594 − 0.215089i \(-0.930996\pi\)
0.976594 − 0.215089i \(-0.0690042\pi\)
\(32\) 46080.0 1.40625
\(33\) − 17931.9i − 0.498982i
\(34\) 36227.6i 0.921727i
\(35\) 0 0
\(36\) 46560.0 0.997942
\(37\) −61577.0 −1.21566 −0.607832 − 0.794066i \(-0.707960\pi\)
−0.607832 + 0.794066i \(0.707960\pi\)
\(38\) 82494.1i 1.50339i
\(39\) −5880.00 −0.0991251
\(40\) 34918.1i 0.545596i
\(41\) 110574.i 1.60436i 0.597082 + 0.802180i \(0.296327\pi\)
−0.597082 + 0.802180i \(0.703673\pi\)
\(42\) 0 0
\(43\) −17414.0 −0.219025 −0.109512 − 0.993985i \(-0.534929\pi\)
−0.109512 + 0.993985i \(0.534929\pi\)
\(44\) 118320. 1.38899
\(45\) − 105846.i − 1.16154i
\(46\) 70956.0 0.728980
\(47\) 30662.5i 0.295334i 0.989037 + 0.147667i \(0.0471764\pi\)
−0.989037 + 0.147667i \(0.952824\pi\)
\(48\) 34142.2i 0.308722i
\(49\) 0 0
\(50\) 209400. 1.67520
\(51\) −36603.0 −0.275935
\(52\) − 38797.9i − 0.275930i
\(53\) −60513.0 −0.406463 −0.203232 − 0.979131i \(-0.565144\pi\)
−0.203232 + 0.979131i \(0.565144\pi\)
\(54\) 190740.i 1.21133i
\(55\) − 268979.i − 1.61670i
\(56\) 0 0
\(57\) −83349.0 −0.450066
\(58\) −47736.0 −0.244659
\(59\) − 215729.i − 1.05039i −0.850981 − 0.525196i \(-0.823992\pi\)
0.850981 − 0.525196i \(-0.176008\pi\)
\(60\) −176400. −0.816667
\(61\) − 162745.i − 0.716999i −0.933530 − 0.358500i \(-0.883288\pi\)
0.933530 − 0.358500i \(-0.116712\pi\)
\(62\) 153785.i 0.645268i
\(63\) 0 0
\(64\) −372736. −1.42188
\(65\) −88200.0 −0.321165
\(66\) 215183.i 0.748473i
\(67\) −268777. −0.893650 −0.446825 − 0.894621i \(-0.647445\pi\)
−0.446825 + 0.894621i \(0.647445\pi\)
\(68\) − 241517.i − 0.768106i
\(69\) 71691.3i 0.218232i
\(70\) 0 0
\(71\) 101922. 0.284769 0.142385 − 0.989811i \(-0.454523\pi\)
0.142385 + 0.989811i \(0.454523\pi\)
\(72\) −111744. −0.299383
\(73\) 317646.i 0.816535i 0.912862 + 0.408267i \(0.133867\pi\)
−0.912862 + 0.408267i \(0.866133\pi\)
\(74\) 738924. 1.82350
\(75\) 211570.i 0.501499i
\(76\) − 549961.i − 1.25283i
\(77\) 0 0
\(78\) 70560.0 0.148688
\(79\) 362231. 0.734690 0.367345 − 0.930085i \(-0.380267\pi\)
0.367345 + 0.930085i \(0.380267\pi\)
\(80\) 512133.i 1.00026i
\(81\) 231561. 0.435723
\(82\) − 1.32689e6i − 2.40654i
\(83\) − 216783.i − 0.379133i −0.981868 − 0.189567i \(-0.939292\pi\)
0.981868 − 0.189567i \(-0.0607083\pi\)
\(84\) 0 0
\(85\) −549045. −0.894028
\(86\) 208968. 0.328537
\(87\) − 48230.7i − 0.0732429i
\(88\) −283968. −0.416698
\(89\) 1.33456e6i 1.89308i 0.322583 + 0.946541i \(0.395449\pi\)
−0.322583 + 0.946541i \(0.604551\pi\)
\(90\) 1.27015e6i 1.74231i
\(91\) 0 0
\(92\) −473040. −0.607483
\(93\) −155379. −0.193172
\(94\) − 367950.i − 0.443001i
\(95\) −1.25024e6 −1.45821
\(96\) − 558690.i − 0.631477i
\(97\) − 1.51409e6i − 1.65896i −0.558535 − 0.829481i \(-0.688636\pi\)
0.558535 − 0.829481i \(-0.311364\pi\)
\(98\) 0 0
\(99\) 860778. 0.887127
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.b.a.48.1 2
3.2 odd 2 441.7.d.a.244.2 2
7.2 even 3 49.7.d.b.31.1 2
7.3 odd 6 49.7.d.b.19.1 2
7.4 even 3 7.7.d.a.5.1 yes 2
7.5 odd 6 7.7.d.a.3.1 ✓ 2
7.6 odd 2 inner 49.7.b.a.48.2 2
21.5 even 6 63.7.m.a.10.1 2
21.11 odd 6 63.7.m.a.19.1 2
21.20 even 2 441.7.d.a.244.1 2
28.11 odd 6 112.7.s.a.33.1 2
28.19 even 6 112.7.s.a.17.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.7.d.a.3.1 ✓ 2 7.5 odd 6
7.7.d.a.5.1 yes 2 7.4 even 3
49.7.b.a.48.1 2 1.1 even 1 trivial
49.7.b.a.48.2 2 7.6 odd 2 inner
49.7.d.b.19.1 2 7.3 odd 6
49.7.d.b.31.1 2 7.2 even 3
63.7.m.a.10.1 2 21.5 even 6
63.7.m.a.19.1 2 21.11 odd 6
112.7.s.a.17.1 2 28.19 even 6
112.7.s.a.33.1 2 28.11 odd 6
441.7.d.a.244.1 2 21.20 even 2
441.7.d.a.244.2 2 3.2 odd 2