Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [495,6,Mod(1,495)] 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("495.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 495.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,7,0,41] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(79.3899908074\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.21865.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 30x + 40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(5.25849\) of defining polynomial
Character \(\chi\) \(=\) 495.1

$q$-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)
 
\(f(q)\) \(=\) \(q+8.45512 q^{2} +39.4891 q^{4} -25.0000 q^{5} +22.2927 q^{7} +63.3212 q^{8} -211.378 q^{10} -121.000 q^{11} -225.547 q^{13} +188.488 q^{14} -728.262 q^{16} +1059.59 q^{17} +2525.94 q^{19} -987.227 q^{20} -1023.07 q^{22} +337.423 q^{23} +625.000 q^{25} -1907.03 q^{26} +880.320 q^{28} +7644.60 q^{29} +6754.80 q^{31} -8183.83 q^{32} +8958.95 q^{34} -557.318 q^{35} -5663.43 q^{37} +21357.2 q^{38} -1583.03 q^{40} +13317.4 q^{41} +9007.03 q^{43} -4778.18 q^{44} +2852.96 q^{46} +16644.2 q^{47} -16310.0 q^{49} +5284.45 q^{50} -8906.65 q^{52} -21261.8 q^{53} +3025.00 q^{55} +1411.60 q^{56} +64636.1 q^{58} +44329.0 q^{59} -36591.1 q^{61} +57112.7 q^{62} -45890.9 q^{64} +5638.68 q^{65} -45840.6 q^{67} +41842.2 q^{68} -4712.19 q^{70} +31877.6 q^{71} +49936.7 q^{73} -47885.0 q^{74} +99747.3 q^{76} -2697.42 q^{77} +48257.1 q^{79} +18206.6 q^{80} +112600. q^{82} -66052.0 q^{83} -26489.7 q^{85} +76155.6 q^{86} -7661.87 q^{88} +124047. q^{89} -5028.06 q^{91} +13324.5 q^{92} +140729. q^{94} -63148.6 q^{95} +73854.3 q^{97} -137903. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 7 q^{2} + 41 q^{4} - 75 q^{5} - 102 q^{7} + 15 q^{8} - 175 q^{10} - 363 q^{11} - 1646 q^{13} + 963 q^{14} - 2687 q^{16} + 1742 q^{17} - 10 q^{19} - 1025 q^{20} - 847 q^{22} + 3876 q^{23} + 1875 q^{25}+ \cdots - 209376 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.45512 1.49467 0.747334 − 0.664448i \(-0.231333\pi\)
0.747334 + 0.664448i \(0.231333\pi\)
\(3\) 0 0
\(4\) 39.4891 1.23403
\(5\) −25.0000 −0.447214
\(6\) 0 0
\(7\) 22.2927 0.171956 0.0859782 − 0.996297i \(-0.472598\pi\)
0.0859782 + 0.996297i \(0.472598\pi\)
\(8\) 63.3212 0.349804
\(9\) 0 0
\(10\) −211.378 −0.668436
\(11\) −121.000 −0.301511
\(12\) 0 0
\(13\) −225.547 −0.370151 −0.185075 − 0.982724i \(-0.559253\pi\)
−0.185075 + 0.982724i \(0.559253\pi\)
\(14\) 188.488 0.257018
\(15\) 0 0
\(16\) −728.262 −0.711194
\(17\) 1059.59 0.889232 0.444616 − 0.895721i \(-0.353340\pi\)
0.444616 + 0.895721i \(0.353340\pi\)
\(18\) 0 0
\(19\) 2525.94 1.60524 0.802620 − 0.596491i \(-0.203439\pi\)
0.802620 + 0.596491i \(0.203439\pi\)
\(20\) −987.227 −0.551877
\(21\) 0 0
\(22\) −1023.07 −0.450660
\(23\) 337.423 0.133001 0.0665006 − 0.997786i \(-0.478817\pi\)
0.0665006 + 0.997786i \(0.478817\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) −1907.03 −0.553253
\(27\) 0 0
\(28\) 880.320 0.212200
\(29\) 7644.60 1.68795 0.843976 − 0.536381i \(-0.180209\pi\)
0.843976 + 0.536381i \(0.180209\pi\)
\(30\) 0 0
\(31\) 6754.80 1.26243 0.631217 − 0.775607i \(-0.282556\pi\)
0.631217 + 0.775607i \(0.282556\pi\)
\(32\) −8183.83 −1.41280
\(33\) 0 0
\(34\) 8958.95 1.32911
\(35\) −557.318 −0.0769012
\(36\) 0 0
\(37\) −5663.43 −0.680104 −0.340052 − 0.940407i \(-0.610445\pi\)
−0.340052 + 0.940407i \(0.610445\pi\)
\(38\) 21357.2 2.39930
\(39\) 0 0
\(40\) −1583.03 −0.156437
\(41\) 13317.4 1.23725 0.618627 − 0.785685i \(-0.287689\pi\)
0.618627 + 0.785685i \(0.287689\pi\)
\(42\) 0 0
\(43\) 9007.03 0.742866 0.371433 − 0.928460i \(-0.378867\pi\)
0.371433 + 0.928460i \(0.378867\pi\)
\(44\) −4778.18 −0.372075
\(45\) 0 0
\(46\) 2852.96 0.198793
\(47\) 16644.2 1.09905 0.549526 − 0.835477i \(-0.314808\pi\)
0.549526 + 0.835477i \(0.314808\pi\)
\(48\) 0 0
\(49\) −16310.0 −0.970431
\(50\) 5284.45 0.298934
\(51\) 0 0
\(52\) −8906.65 −0.456779
\(53\) −21261.8 −1.03971 −0.519853 − 0.854256i \(-0.674013\pi\)
−0.519853 + 0.854256i \(0.674013\pi\)
\(54\) 0 0
\(55\) 3025.00 0.134840
\(56\) 1411.60 0.0601509
\(57\) 0 0
\(58\) 64636.1 2.52293
\(59\) 44329.0 1.65790 0.828948 − 0.559325i \(-0.188940\pi\)
0.828948 + 0.559325i \(0.188940\pi\)
\(60\) 0 0
\(61\) −36591.1 −1.25907 −0.629537 − 0.776970i \(-0.716755\pi\)
−0.629537 + 0.776970i \(0.716755\pi\)
\(62\) 57112.7 1.88692
\(63\) 0 0
\(64\) −45890.9 −1.40048
\(65\) 5638.68 0.165537
\(66\) 0 0
\(67\) −45840.6 −1.24757 −0.623783 − 0.781598i \(-0.714405\pi\)
−0.623783 + 0.781598i \(0.714405\pi\)
\(68\) 41842.2 1.09734
\(69\) 0 0
\(70\) −4712.19 −0.114942
\(71\) 31877.6 0.750481 0.375240 − 0.926928i \(-0.377560\pi\)
0.375240 + 0.926928i \(0.377560\pi\)
\(72\) 0 0
\(73\) 49936.7 1.09676 0.548382 − 0.836228i \(-0.315244\pi\)
0.548382 + 0.836228i \(0.315244\pi\)
\(74\) −47885.0 −1.01653
\(75\) 0 0
\(76\) 99747.3 1.98092
\(77\) −2697.42 −0.0518468
\(78\) 0 0
\(79\) 48257.1 0.869949 0.434974 − 0.900443i \(-0.356757\pi\)
0.434974 + 0.900443i \(0.356757\pi\)
\(80\) 18206.6 0.318056
\(81\) 0 0
\(82\) 112600. 1.84928
\(83\) −66052.0 −1.05242 −0.526212 − 0.850353i \(-0.676388\pi\)
−0.526212 + 0.850353i \(0.676388\pi\)
\(84\) 0 0
\(85\) −26489.7 −0.397677
\(86\) 76155.6 1.11034
\(87\) 0 0
\(88\) −7661.87 −0.105470
\(89\) 124047. 1.66002 0.830009 − 0.557750i \(-0.188335\pi\)
0.830009 + 0.557750i \(0.188335\pi\)
\(90\) 0 0
\(91\) −5028.06 −0.0636498
\(92\) 13324.5 0.164128
\(93\) 0 0
\(94\) 140729. 1.64272
\(95\) −63148.6 −0.717885
\(96\) 0 0
\(97\) 73854.3 0.796978 0.398489 − 0.917173i \(-0.369535\pi\)
0.398489 + 0.917173i \(0.369535\pi\)
\(98\) −137903. −1.45047
\(99\) 0 0
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 495.6.a.f.1.3 3
3.2 odd 2 55.6.a.a.1.1 ✓ 3
12.11 even 2 880.6.a.l.1.3 3
15.2 even 4 275.6.b.c.199.1 6
15.8 even 4 275.6.b.c.199.6 6
15.14 odd 2 275.6.a.c.1.3 3
33.32 even 2 605.6.a.b.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.a.1.1 ✓ 3 3.2 odd 2
275.6.a.c.1.3 3 15.14 odd 2
275.6.b.c.199.1 6 15.2 even 4
275.6.b.c.199.6 6 15.8 even 4
495.6.a.f.1.3 3 1.1 even 1 trivial
605.6.a.b.1.3 3 33.32 even 2
880.6.a.l.1.3 3 12.11 even 2