Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,10,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, 10); 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, 10, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-6,86] 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: \(25.2367559720\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-6.44622\) 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)
 
\(f(q)\) \(=\) \(q+10.8924 q^{2} +195.817 q^{3} -393.355 q^{4} -200.782 q^{5} +2132.92 q^{6} -9861.52 q^{8} +18661.3 q^{9} -2187.01 q^{10} +63864.3 q^{11} -77025.5 q^{12} +164679. q^{13} -39316.5 q^{15} +93981.5 q^{16} +362910. q^{17} +203267. q^{18} +436498. q^{19} +78978.6 q^{20} +695638. q^{22} +918199. q^{23} -1.93105e6 q^{24} -1.91281e6 q^{25} +1.79375e6 q^{26} -200076. q^{27} -3.68643e6 q^{29} -428253. q^{30} -3.47629e6 q^{31} +6.07279e6 q^{32} +1.25057e7 q^{33} +3.95298e6 q^{34} -7.34049e6 q^{36} +1.88149e7 q^{37} +4.75453e6 q^{38} +3.22469e7 q^{39} +1.98002e6 q^{40} -2.40714e6 q^{41} -1.25306e7 q^{43} -2.51213e7 q^{44} -3.74685e6 q^{45} +1.00014e7 q^{46} +5.54509e7 q^{47} +1.84032e7 q^{48} -2.08352e7 q^{50} +7.10639e7 q^{51} -6.47772e7 q^{52} -9.26889e7 q^{53} -2.17931e6 q^{54} -1.28228e7 q^{55} +8.54737e7 q^{57} -4.01542e7 q^{58} +2.52600e7 q^{59} +1.54653e7 q^{60} -6.93275e7 q^{61} -3.78653e7 q^{62} +1.80290e7 q^{64} -3.30646e7 q^{65} +1.36218e8 q^{66} -2.33494e7 q^{67} -1.42752e8 q^{68} +1.79799e8 q^{69} -1.06194e8 q^{71} -1.84028e8 q^{72} +2.10115e8 q^{73} +2.04940e8 q^{74} -3.74561e8 q^{75} -1.71699e8 q^{76} +3.51247e8 q^{78} -149606. q^{79} -1.88698e7 q^{80} -4.06488e8 q^{81} -2.62197e7 q^{82} -5.21565e8 q^{83} -7.28659e7 q^{85} -1.36489e8 q^{86} -7.21865e8 q^{87} -6.29799e8 q^{88} -2.98587e8 q^{89} -4.08123e7 q^{90} -3.61178e8 q^{92} -6.80716e8 q^{93} +6.03996e8 q^{94} -8.76410e7 q^{95} +1.18915e9 q^{96} +8.95983e8 q^{97} +1.19179e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{2} + 86 q^{3} - 620 q^{4} + 2238 q^{5} + 3988 q^{6} + 2616 q^{8} + 11038 q^{9} - 43384 q^{10} + 35316 q^{11} - 52136 q^{12} + 26530 q^{13} - 307136 q^{15} - 752 q^{16} + 463920 q^{17} + 332042 q^{18}+ \cdots + 1409417860 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.8924 0.481383 0.240691 − 0.970602i \(-0.422626\pi\)
0.240691 + 0.970602i \(0.422626\pi\)
\(3\) 195.817 1.39574 0.697870 − 0.716225i \(-0.254131\pi\)
0.697870 + 0.716225i \(0.254131\pi\)
\(4\) −393.355 −0.768271
\(5\) −200.782 −0.143668 −0.0718340 − 0.997417i \(-0.522885\pi\)
−0.0718340 + 0.997417i \(0.522885\pi\)
\(6\) 2132.92 0.671885
\(7\) 0 0
\(8\) −9861.52 −0.851215
\(9\) 18661.3 0.948090
\(10\) −2187.01 −0.0691593
\(11\) 63864.3 1.31520 0.657599 − 0.753369i \(-0.271572\pi\)
0.657599 + 0.753369i \(0.271572\pi\)
\(12\) −77025.5 −1.07231
\(13\) 164679. 1.59916 0.799581 − 0.600558i \(-0.205055\pi\)
0.799581 + 0.600558i \(0.205055\pi\)
\(14\) 0 0
\(15\) −39316.5 −0.200523
\(16\) 93981.5 0.358511
\(17\) 362910. 1.05385 0.526925 − 0.849912i \(-0.323345\pi\)
0.526925 + 0.849912i \(0.323345\pi\)
\(18\) 203267. 0.456394
\(19\) 436498. 0.768406 0.384203 − 0.923249i \(-0.374476\pi\)
0.384203 + 0.923249i \(0.374476\pi\)
\(20\) 78978.6 0.110376
\(21\) 0 0
\(22\) 695638. 0.633113
\(23\) 918199. 0.684166 0.342083 − 0.939670i \(-0.388868\pi\)
0.342083 + 0.939670i \(0.388868\pi\)
\(24\) −1.93105e6 −1.18807
\(25\) −1.91281e6 −0.979359
\(26\) 1.79375e6 0.769809
\(27\) −200076. −0.0724531
\(28\) 0 0
\(29\) −3.68643e6 −0.967865 −0.483932 − 0.875105i \(-0.660792\pi\)
−0.483932 + 0.875105i \(0.660792\pi\)
\(30\) −428253. −0.0965284
\(31\) −3.47629e6 −0.676064 −0.338032 − 0.941135i \(-0.609761\pi\)
−0.338032 + 0.941135i \(0.609761\pi\)
\(32\) 6.07279e6 1.02380
\(33\) 1.25057e7 1.83567
\(34\) 3.95298e6 0.507305
\(35\) 0 0
\(36\) −7.34049e6 −0.728390
\(37\) 1.88149e7 1.65042 0.825210 − 0.564826i \(-0.191057\pi\)
0.825210 + 0.564826i \(0.191057\pi\)
\(38\) 4.75453e6 0.369897
\(39\) 3.22469e7 2.23201
\(40\) 1.98002e6 0.122292
\(41\) −2.40714e6 −0.133038 −0.0665188 − 0.997785i \(-0.521189\pi\)
−0.0665188 + 0.997785i \(0.521189\pi\)
\(42\) 0 0
\(43\) −1.25306e7 −0.558938 −0.279469 − 0.960155i \(-0.590158\pi\)
−0.279469 + 0.960155i \(0.590158\pi\)
\(44\) −2.51213e7 −1.01043
\(45\) −3.74685e6 −0.136210
\(46\) 1.00014e7 0.329346
\(47\) 5.54509e7 1.65756 0.828779 − 0.559577i \(-0.189036\pi\)
0.828779 + 0.559577i \(0.189036\pi\)
\(48\) 1.84032e7 0.500388
\(49\) 0 0
\(50\) −2.08352e7 −0.471447
\(51\) 7.10639e7 1.47090
\(52\) −6.47772e7 −1.22859
\(53\) −9.26889e7 −1.61356 −0.806782 − 0.590849i \(-0.798793\pi\)
−0.806782 + 0.590849i \(0.798793\pi\)
\(54\) −2.17931e6 −0.0348777
\(55\) −1.28228e7 −0.188952
\(56\) 0 0
\(57\) 8.54737e7 1.07250
\(58\) −4.01542e7 −0.465913
\(59\) 2.52600e7 0.271393 0.135696 − 0.990750i \(-0.456673\pi\)
0.135696 + 0.990750i \(0.456673\pi\)
\(60\) 1.54653e7 0.154056
\(61\) −6.93275e7 −0.641093 −0.320547 − 0.947233i \(-0.603867\pi\)
−0.320547 + 0.947233i \(0.603867\pi\)
\(62\) −3.78653e7 −0.325446
\(63\) 0 0
\(64\) 1.80290e7 0.134326
\(65\) −3.30646e7 −0.229748
\(66\) 1.36218e8 0.883661
\(67\) −2.33494e7 −0.141559 −0.0707796 − 0.997492i \(-0.522549\pi\)
−0.0707796 + 0.997492i \(0.522549\pi\)
\(68\) −1.42752e8 −0.809643
\(69\) 1.79799e8 0.954918
\(70\) 0 0
\(71\) −1.06194e8 −0.495950 −0.247975 − 0.968766i \(-0.579765\pi\)
−0.247975 + 0.968766i \(0.579765\pi\)
\(72\) −1.84028e8 −0.807028
\(73\) 2.10115e8 0.865974 0.432987 − 0.901400i \(-0.357460\pi\)
0.432987 + 0.901400i \(0.357460\pi\)
\(74\) 2.04940e8 0.794483
\(75\) −3.74561e8 −1.36693
\(76\) −1.71699e8 −0.590344
\(77\) 0 0
\(78\) 3.51247e8 1.07445
\(79\) −149606. −0.000432144 0 −0.000216072 − 1.00000i \(-0.500069\pi\)
−0.000216072 1.00000i \(0.500069\pi\)
\(80\) −1.88698e7 −0.0515066
\(81\) −4.06488e8 −1.04922
\(82\) −2.62197e7 −0.0640420
\(83\) −5.21565e8 −1.20630 −0.603152 − 0.797626i \(-0.706089\pi\)
−0.603152 + 0.797626i \(0.706089\pi\)
\(84\) 0 0
\(85\) −7.28659e7 −0.151405
\(86\) −1.36489e8 −0.269063
\(87\) −7.21865e8 −1.35089
\(88\) −6.29799e8 −1.11952
\(89\) −2.98587e8 −0.504448 −0.252224 − 0.967669i \(-0.581162\pi\)
−0.252224 + 0.967669i \(0.581162\pi\)
\(90\) −4.08123e7 −0.0655692
\(91\) 0 0
\(92\) −3.61178e8 −0.525625
\(93\) −6.80716e8 −0.943610
\(94\) 6.03996e8 0.797919
\(95\) −8.76410e7 −0.110395
\(96\) 1.18915e9 1.42895
\(97\) 8.95983e8 1.02761 0.513803 − 0.857908i \(-0.328236\pi\)
0.513803 + 0.857908i \(0.328236\pi\)
\(98\) 0 0
\(99\) 1.19179e9 1.24693
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 49.10.a.b.1.2 2
7.2 even 3 49.10.c.b.18.1 4
7.3 odd 6 49.10.c.c.30.1 4
7.4 even 3 49.10.c.b.30.1 4
7.5 odd 6 49.10.c.c.18.1 4
7.6 odd 2 7.10.a.a.1.2 ✓ 2
21.20 even 2 63.10.a.d.1.1 2
28.27 even 2 112.10.a.e.1.2 2
35.13 even 4 175.10.b.b.99.2 4
35.27 even 4 175.10.b.b.99.3 4
35.34 odd 2 175.10.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.10.a.a.1.2 ✓ 2 7.6 odd 2
49.10.a.b.1.2 2 1.1 even 1 trivial
49.10.c.b.18.1 4 7.2 even 3
49.10.c.b.30.1 4 7.4 even 3
49.10.c.c.18.1 4 7.5 odd 6
49.10.c.c.30.1 4 7.3 odd 6
63.10.a.d.1.1 2 21.20 even 2
112.10.a.e.1.2 2 28.27 even 2
175.10.a.b.1.1 2 35.34 odd 2
175.10.b.b.99.2 4 35.13 even 4
175.10.b.b.99.3 4 35.27 even 4