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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 = [4] 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: \(4\)
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: \( 2^{2}\cdot 3\cdot 7 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 48.3
Root \(0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 49.48
Dual form 49.7.b.b.48.4

$q$-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)
 
\(f(q)\) \(=\) \(q+8.24264 q^{2} -26.6472i q^{3} +3.94113 q^{4} -79.1732i q^{5} -219.643i q^{6} -495.044 q^{8} +18.9260 q^{9} -652.596i q^{10} -1708.92 q^{11} -105.020i q^{12} -3129.09i q^{13} -2109.75 q^{15} -4332.70 q^{16} +4076.05i q^{17} +156.000 q^{18} -5872.68i q^{19} -312.032i q^{20} -14086.0 q^{22} +13321.5 q^{23} +13191.5i q^{24} +9356.60 q^{25} -25792.0i q^{26} -19930.1i q^{27} +6510.23 q^{29} -17389.9 q^{30} -11993.4i q^{31} -4030.09 q^{32} +45537.9i q^{33} +33597.4i q^{34} +74.5896 q^{36} -4640.90 q^{37} -48406.4i q^{38} -83381.6 q^{39} +39194.2i q^{40} -19308.8i q^{41} +91636.4 q^{43} -6735.06 q^{44} -1498.43i q^{45} +109804. q^{46} +64432.5i q^{47} +115454. i q^{48} +77123.1 q^{50} +108615. q^{51} -12332.1i q^{52} +149600. q^{53} -164277. i q^{54} +135301. i q^{55} -156491. q^{57} +53661.5 q^{58} +61031.7i q^{59} -8314.77 q^{60} +98615.3i q^{61} -98857.1i q^{62} +244074. q^{64} -247740. q^{65} +375353. i q^{66} -311812. q^{67} +16064.2i q^{68} -354981. i q^{69} -401209. q^{71} -9369.18 q^{72} -672407. i q^{73} -38253.3 q^{74} -249327. i q^{75} -23145.0i q^{76} -687285. q^{78} -320152. q^{79} +343034. i q^{80} -517286. q^{81} -159155. i q^{82} -832356. i q^{83} +322714. q^{85} +755326. q^{86} -173479. i q^{87} +845989. q^{88} -379497. i q^{89} -12351.0i q^{90} +52501.7 q^{92} -319590. q^{93} +531094. i q^{94} -464959. q^{95} +107391. i q^{96} +1.05514e6i q^{97} -32342.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{2} - 120 q^{4} - 928 q^{8} - 1248 q^{9} - 3764 q^{11} - 16500 q^{15} - 496 q^{16} + 624 q^{18} - 28088 q^{22} - 4940 q^{23} - 5000 q^{25} - 34544 q^{29} - 31800 q^{30} - 14016 q^{32} + 82368 q^{36}+ \cdots + 157872 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.24264 1.03033 0.515165 − 0.857091i \(-0.327731\pi\)
0.515165 + 0.857091i \(0.327731\pi\)
\(3\) − 26.6472i − 0.986934i −0.869765 − 0.493467i \(-0.835729\pi\)
0.869765 − 0.493467i \(-0.164271\pi\)
\(4\) 3.94113 0.0615801
\(5\) − 79.1732i − 0.633386i −0.948528 − 0.316693i \(-0.897428\pi\)
0.948528 − 0.316693i \(-0.102572\pi\)
\(6\) − 219.643i − 1.01687i
\(7\) 0 0
\(8\) −495.044 −0.966882
\(9\) 18.9260 0.0259616
\(10\) − 652.596i − 0.652596i
\(11\) −1708.92 −1.28394 −0.641968 − 0.766732i \(-0.721882\pi\)
−0.641968 + 0.766732i \(0.721882\pi\)
\(12\) − 105.020i − 0.0607755i
\(13\) − 3129.09i − 1.42426i −0.702049 − 0.712129i \(-0.747731\pi\)
0.702049 − 0.712129i \(-0.252269\pi\)
\(14\) 0 0
\(15\) −2109.75 −0.625110
\(16\) −4332.70 −1.05779
\(17\) 4076.05i 0.829646i 0.909902 + 0.414823i \(0.136156\pi\)
−0.909902 + 0.414823i \(0.863844\pi\)
\(18\) 156.000 0.0267490
\(19\) − 5872.68i − 0.856201i −0.903731 − 0.428101i \(-0.859183\pi\)
0.903731 − 0.428101i \(-0.140817\pi\)
\(20\) − 312.032i − 0.0390039i
\(21\) 0 0
\(22\) −14086.0 −1.32288
\(23\) 13321.5 1.09489 0.547444 − 0.836842i \(-0.315601\pi\)
0.547444 + 0.836842i \(0.315601\pi\)
\(24\) 13191.5i 0.954249i
\(25\) 9356.60 0.598823
\(26\) − 25792.0i − 1.46746i
\(27\) − 19930.1i − 1.01256i
\(28\) 0 0
\(29\) 6510.23 0.266933 0.133466 − 0.991053i \(-0.457389\pi\)
0.133466 + 0.991053i \(0.457389\pi\)
\(30\) −17389.9 −0.644069
\(31\) − 11993.4i − 0.402584i −0.979531 − 0.201292i \(-0.935486\pi\)
0.979531 − 0.201292i \(-0.0645140\pi\)
\(32\) −4030.09 −0.122989
\(33\) 45537.9i 1.26716i
\(34\) 33597.4i 0.854809i
\(35\) 0 0
\(36\) 74.5896 0.00159871
\(37\) −4640.90 −0.0916214 −0.0458107 − 0.998950i \(-0.514587\pi\)
−0.0458107 + 0.998950i \(0.514587\pi\)
\(38\) − 48406.4i − 0.882170i
\(39\) −83381.6 −1.40565
\(40\) 39194.2i 0.612409i
\(41\) − 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 \(0.304504\pi\)
\(44\) −6735.06 −0.0790648
\(45\) − 1498.43i − 0.0164437i
\(46\) 109804. 1.12810
\(47\) 64432.5i 0.620599i 0.950639 + 0.310300i \(0.100429\pi\)
−0.950639 + 0.310300i \(0.899571\pi\)
\(48\) 115454.i 1.04397i
\(49\) 0 0
\(50\) 77123.1 0.616985
\(51\) 108615. 0.818806
\(52\) − 12332.1i − 0.0877059i
\(53\) 149600. 1.00486 0.502428 − 0.864619i \(-0.332440\pi\)
0.502428 + 0.864619i \(0.332440\pi\)
\(54\) − 164277.i − 1.04327i
\(55\) 135301.i 0.813226i
\(56\) 0 0
\(57\) −156491. −0.845014
\(58\) 53661.5 0.275029
\(59\) 61031.7i 0.297166i 0.988900 + 0.148583i \(0.0474713\pi\)
−0.988900 + 0.148583i \(0.952529\pi\)
\(60\) −8314.77 −0.0384943
\(61\) 98615.3i 0.434465i 0.976120 + 0.217233i \(0.0697030\pi\)
−0.976120 + 0.217233i \(0.930297\pi\)
\(62\) − 98857.1i − 0.414794i
\(63\) 0 0
\(64\) 244074. 0.931069
\(65\) −247740. −0.902104
\(66\) 375353.i 1.30559i
\(67\) −311812. −1.03674 −0.518369 − 0.855157i \(-0.673461\pi\)
−0.518369 + 0.855157i \(0.673461\pi\)
\(68\) 16064.2i 0.0510897i
\(69\) − 354981.i − 1.08058i
\(70\) 0 0
\(71\) −401209. −1.12097 −0.560487 − 0.828163i \(-0.689386\pi\)
−0.560487 + 0.828163i \(0.689386\pi\)
\(72\) −9369.18 −0.0251018
\(73\) − 672407.i − 1.72848i −0.503081 − 0.864239i \(-0.667800\pi\)
0.503081 − 0.864239i \(-0.332200\pi\)
\(74\) −38253.3 −0.0944003
\(75\) − 249327.i − 0.590998i
\(76\) − 23145.0i − 0.0527249i
\(77\) 0 0
\(78\) −687285. −1.44828
\(79\) −320152. −0.649344 −0.324672 − 0.945827i \(-0.605254\pi\)
−0.324672 + 0.945827i \(0.605254\pi\)
\(80\) 343034.i 0.669988i
\(81\) −517286. −0.973364
\(82\) − 159155.i − 0.288655i
\(83\) − 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
\(86\) 755326. 1.18751
\(87\) − 173479.i − 0.263445i
\(88\) 845989. 1.24141
\(89\) − 379497.i − 0.538317i −0.963096 − 0.269158i \(-0.913254\pi\)
0.963096 − 0.269158i \(-0.0867455\pi\)
\(90\) − 12351.0i − 0.0169424i
\(91\) 0 0
\(92\) 52501.7 0.0674233
\(93\) −319590. −0.397324
\(94\) 531094.i 0.639422i
\(95\) −464959. −0.542306
\(96\) 107391.i 0.121382i
\(97\) 1.05514e6i 1.15610i 0.816001 + 0.578050i \(0.196186\pi\)
−0.816001 + 0.578050i \(0.803814\pi\)
\(98\) 0 0
\(99\) −32342.9 −0.0333330
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.b.48.3 4
3.2 odd 2 441.7.d.b.244.2 4
7.2 even 3 49.7.d.c.31.1 4
7.3 odd 6 49.7.d.c.19.1 4
7.4 even 3 7.7.d.b.5.1 yes 4
7.5 odd 6 7.7.d.b.3.1 ✓ 4
7.6 odd 2 inner 49.7.b.b.48.4 4
21.5 even 6 63.7.m.b.10.2 4
21.11 odd 6 63.7.m.b.19.2 4
21.20 even 2 441.7.d.b.244.1 4
28.11 odd 6 112.7.s.b.33.2 4
28.19 even 6 112.7.s.b.17.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.7.d.b.3.1 ✓ 4 7.5 odd 6
7.7.d.b.5.1 yes 4 7.4 even 3
49.7.b.b.48.3 4 1.1 even 1 trivial
49.7.b.b.48.4 4 7.6 odd 2 inner
49.7.d.c.19.1 4 7.3 odd 6
49.7.d.c.31.1 4 7.2 even 3
63.7.m.b.10.2 4 21.5 even 6
63.7.m.b.19.2 4 21.11 odd 6
112.7.s.b.17.2 4 28.19 even 6
112.7.s.b.33.2 4 28.11 odd 6
441.7.d.b.244.1 4 21.20 even 2
441.7.d.b.244.2 4 3.2 odd 2