Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,6,Mod(1,49)] 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("49.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-10,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.85880717084\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{113})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 59x^{2} + 60x + 674 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-6.22929\) of defining polynomial
Character \(\chi\) \(=\) 49.1

$q$-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)
 
\(f(q)\) \(=\) \(q-7.81507 q^{2} +23.5186 q^{3} +29.0754 q^{4} -74.2753 q^{5} -183.799 q^{6} +22.8562 q^{8} +310.123 q^{9} +580.467 q^{10} -424.219 q^{11} +683.811 q^{12} -252.233 q^{13} -1746.85 q^{15} -1109.03 q^{16} -1104.35 q^{17} -2423.64 q^{18} -6.47100 q^{19} -2159.58 q^{20} +3315.30 q^{22} -3612.39 q^{23} +537.546 q^{24} +2391.82 q^{25} +1971.22 q^{26} +1578.65 q^{27} -5005.02 q^{29} +13651.8 q^{30} +2821.69 q^{31} +7935.79 q^{32} -9977.03 q^{33} +8630.55 q^{34} +9016.95 q^{36} -2046.88 q^{37} +50.5713 q^{38} -5932.17 q^{39} -1697.65 q^{40} +9393.81 q^{41} +10320.8 q^{43} -12334.3 q^{44} -23034.5 q^{45} +28231.1 q^{46} +17035.6 q^{47} -26082.9 q^{48} -18692.3 q^{50} -25972.7 q^{51} -7333.78 q^{52} -39506.7 q^{53} -12337.3 q^{54} +31509.0 q^{55} -152.189 q^{57} +39114.6 q^{58} +33949.8 q^{59} -50790.3 q^{60} +28295.2 q^{61} -22051.7 q^{62} -26529.7 q^{64} +18734.7 q^{65} +77971.2 q^{66} +56100.9 q^{67} -32109.3 q^{68} -84958.2 q^{69} -15537.4 q^{71} +7088.26 q^{72} -78219.5 q^{73} +15996.5 q^{74} +56252.3 q^{75} -188.147 q^{76} +46360.3 q^{78} -45335.5 q^{79} +82373.9 q^{80} -38232.4 q^{81} -73413.3 q^{82} +1381.82 q^{83} +82025.7 q^{85} -80657.6 q^{86} -117711. q^{87} -9696.05 q^{88} -68879.4 q^{89} +180016. q^{90} -105031. q^{92} +66362.1 q^{93} -133134. q^{94} +480.635 q^{95} +186638. q^{96} +108857. q^{97} -131560. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{2} + 10 q^{4} - 270 q^{8} + 220 q^{9} - 1952 q^{11} - 4096 q^{15} - 1566 q^{16} - 5974 q^{18} + 3524 q^{22} - 7136 q^{23} + 2764 q^{25} - 3352 q^{29} + 25608 q^{30} + 27810 q^{32} + 27670 q^{36}+ \cdots - 42272 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.81507 −1.38152 −0.690761 − 0.723083i \(-0.742724\pi\)
−0.690761 + 0.723083i \(0.742724\pi\)
\(3\) 23.5186 1.50872 0.754359 − 0.656462i \(-0.227948\pi\)
0.754359 + 0.656462i \(0.227948\pi\)
\(4\) 29.0754 0.908605
\(5\) −74.2753 −1.32868 −0.664339 − 0.747432i \(-0.731287\pi\)
−0.664339 + 0.747432i \(0.731287\pi\)
\(6\) −183.799 −2.08433
\(7\) 0 0
\(8\) 22.8562 0.126264
\(9\) 310.123 1.27623
\(10\) 580.467 1.83560
\(11\) −424.219 −1.05708 −0.528541 − 0.848908i \(-0.677261\pi\)
−0.528541 + 0.848908i \(0.677261\pi\)
\(12\) 683.811 1.37083
\(13\) −252.233 −0.413946 −0.206973 − 0.978347i \(-0.566361\pi\)
−0.206973 + 0.978347i \(0.566361\pi\)
\(14\) 0 0
\(15\) −1746.85 −2.00460
\(16\) −1109.03 −1.08304
\(17\) −1104.35 −0.926794 −0.463397 − 0.886151i \(-0.653370\pi\)
−0.463397 + 0.886151i \(0.653370\pi\)
\(18\) −2423.64 −1.76314
\(19\) −6.47100 −0.00411232 −0.00205616 − 0.999998i \(-0.500654\pi\)
−0.00205616 + 0.999998i \(0.500654\pi\)
\(20\) −2159.58 −1.20724
\(21\) 0 0
\(22\) 3315.30 1.46038
\(23\) −3612.39 −1.42388 −0.711942 − 0.702239i \(-0.752184\pi\)
−0.711942 + 0.702239i \(0.752184\pi\)
\(24\) 537.546 0.190497
\(25\) 2391.82 0.765383
\(26\) 1971.22 0.571876
\(27\) 1578.65 0.416751
\(28\) 0 0
\(29\) −5005.02 −1.10512 −0.552561 − 0.833472i \(-0.686350\pi\)
−0.552561 + 0.833472i \(0.686350\pi\)
\(30\) 13651.8 2.76940
\(31\) 2821.69 0.527357 0.263679 − 0.964611i \(-0.415064\pi\)
0.263679 + 0.964611i \(0.415064\pi\)
\(32\) 7935.79 1.36998
\(33\) −9977.03 −1.59484
\(34\) 8630.55 1.28039
\(35\) 0 0
\(36\) 9016.95 1.15959
\(37\) −2046.88 −0.245803 −0.122902 − 0.992419i \(-0.539220\pi\)
−0.122902 + 0.992419i \(0.539220\pi\)
\(38\) 50.5713 0.00568127
\(39\) −5932.17 −0.624528
\(40\) −1697.65 −0.167764
\(41\) 9393.81 0.872734 0.436367 − 0.899769i \(-0.356265\pi\)
0.436367 + 0.899769i \(0.356265\pi\)
\(42\) 0 0
\(43\) 10320.8 0.851218 0.425609 − 0.904907i \(-0.360060\pi\)
0.425609 + 0.904907i \(0.360060\pi\)
\(44\) −12334.3 −0.960470
\(45\) −23034.5 −1.69570
\(46\) 28231.1 1.96713
\(47\) 17035.6 1.12490 0.562448 − 0.826833i \(-0.309860\pi\)
0.562448 + 0.826833i \(0.309860\pi\)
\(48\) −26082.9 −1.63400
\(49\) 0 0
\(50\) −18692.3 −1.05739
\(51\) −25972.7 −1.39827
\(52\) −7333.78 −0.376114
\(53\) −39506.7 −1.93188 −0.965941 − 0.258761i \(-0.916686\pi\)
−0.965941 + 0.258761i \(0.916686\pi\)
\(54\) −12337.3 −0.575750
\(55\) 31509.0 1.40452
\(56\) 0 0
\(57\) −152.189 −0.00620433
\(58\) 39114.6 1.52675
\(59\) 33949.8 1.26972 0.634859 − 0.772628i \(-0.281058\pi\)
0.634859 + 0.772628i \(0.281058\pi\)
\(60\) −50790.3 −1.82139
\(61\) 28295.2 0.973618 0.486809 − 0.873508i \(-0.338161\pi\)
0.486809 + 0.873508i \(0.338161\pi\)
\(62\) −22051.7 −0.728556
\(63\) 0 0
\(64\) −26529.7 −0.809621
\(65\) 18734.7 0.550001
\(66\) 77971.2 2.20330
\(67\) 56100.9 1.52680 0.763402 − 0.645924i \(-0.223528\pi\)
0.763402 + 0.645924i \(0.223528\pi\)
\(68\) −32109.3 −0.842090
\(69\) −84958.2 −2.14824
\(70\) 0 0
\(71\) −15537.4 −0.365791 −0.182895 − 0.983132i \(-0.558547\pi\)
−0.182895 + 0.983132i \(0.558547\pi\)
\(72\) 7088.26 0.161142
\(73\) −78219.5 −1.71794 −0.858970 − 0.512025i \(-0.828895\pi\)
−0.858970 + 0.512025i \(0.828895\pi\)
\(74\) 15996.5 0.339583
\(75\) 56252.3 1.15475
\(76\) −188.147 −0.00373648
\(77\) 0 0
\(78\) 46360.3 0.862800
\(79\) −45335.5 −0.817279 −0.408640 − 0.912696i \(-0.633997\pi\)
−0.408640 + 0.912696i \(0.633997\pi\)
\(80\) 82373.9 1.43901
\(81\) −38232.4 −0.647469
\(82\) −73413.3 −1.20570
\(83\) 1381.82 0.0220169 0.0110085 − 0.999939i \(-0.496496\pi\)
0.0110085 + 0.999939i \(0.496496\pi\)
\(84\) 0 0
\(85\) 82025.7 1.23141
\(86\) −80657.6 −1.17598
\(87\) −117711. −1.66732
\(88\) −9696.05 −0.133471
\(89\) −68879.4 −0.921753 −0.460876 − 0.887464i \(-0.652465\pi\)
−0.460876 + 0.887464i \(0.652465\pi\)
\(90\) 180016. 2.34264
\(91\) 0 0
\(92\) −105031. −1.29375
\(93\) 66362.1 0.795633
\(94\) −133134. −1.55407
\(95\) 480.635 0.00546395
\(96\) 186638. 2.06692
\(97\) 108857. 1.17470 0.587351 − 0.809332i \(-0.300171\pi\)
0.587351 + 0.809332i \(0.300171\pi\)
\(98\) 0 0
\(99\) −131560. −1.34908
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 49.6.a.g.1.2 yes 4
3.2 odd 2 441.6.a.z.1.4 4
4.3 odd 2 784.6.a.bf.1.1 4
7.2 even 3 49.6.c.h.18.3 8
7.3 odd 6 49.6.c.h.30.4 8
7.4 even 3 49.6.c.h.30.3 8
7.5 odd 6 49.6.c.h.18.4 8
7.6 odd 2 inner 49.6.a.g.1.1 ✓ 4
21.20 even 2 441.6.a.z.1.3 4
28.27 even 2 784.6.a.bf.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.g.1.1 ✓ 4 7.6 odd 2 inner
49.6.a.g.1.2 yes 4 1.1 even 1 trivial
49.6.c.h.18.3 8 7.2 even 3
49.6.c.h.18.4 8 7.5 odd 6
49.6.c.h.30.3 8 7.4 even 3
49.6.c.h.30.4 8 7.3 odd 6
441.6.a.z.1.3 4 21.20 even 2
441.6.a.z.1.4 4 3.2 odd 2
784.6.a.bf.1.1 4 4.3 odd 2
784.6.a.bf.1.4 4 28.27 even 2