Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,6,Mod(1,275)] 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("275.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,7,36] 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: \(44.1055504486\)
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: \(\Z[a_1, a_2, a_3]\)
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 \(\chi\) \(=\) 275.1

$q$-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)
 
\(f(q)\) \(=\) \(q+8.45512 q^{2} +26.7755 q^{3} +39.4891 q^{4} +226.390 q^{6} -22.2927 q^{7} +63.3212 q^{8} +473.926 q^{9} +121.000 q^{11} +1057.34 q^{12} +225.547 q^{13} -188.488 q^{14} -728.262 q^{16} +1059.59 q^{17} +4007.11 q^{18} +2525.94 q^{19} -596.899 q^{21} +1023.07 q^{22} +337.423 q^{23} +1695.46 q^{24} +1907.03 q^{26} +6183.16 q^{27} -880.320 q^{28} -7644.60 q^{29} +6754.80 q^{31} -8183.83 q^{32} +3239.83 q^{33} +8958.95 q^{34} +18714.9 q^{36} +5663.43 q^{37} +21357.2 q^{38} +6039.13 q^{39} -13317.4 q^{41} -5046.85 q^{42} -9007.03 q^{43} +4778.18 q^{44} +2852.96 q^{46} +16644.2 q^{47} -19499.6 q^{48} -16310.0 q^{49} +28371.0 q^{51} +8906.65 q^{52} -21261.8 q^{53} +52279.4 q^{54} -1411.60 q^{56} +67633.4 q^{57} -64636.1 q^{58} -44329.0 q^{59} -36591.1 q^{61} +57112.7 q^{62} -10565.1 q^{63} -45890.9 q^{64} +27393.2 q^{66} +45840.6 q^{67} +41842.2 q^{68} +9034.68 q^{69} -31877.6 q^{71} +30009.6 q^{72} -49936.7 q^{73} +47885.0 q^{74} +99747.3 q^{76} -2697.42 q^{77} +51061.6 q^{78} +48257.1 q^{79} +50393.1 q^{81} -112600. q^{82} -66052.0 q^{83} -23571.0 q^{84} -76155.6 q^{86} -204688. q^{87} +7661.87 q^{88} -124047. q^{89} -5028.06 q^{91} +13324.5 q^{92} +180863. q^{93} +140729. q^{94} -219126. q^{96} -73854.3 q^{97} -137903. q^{98} +57345.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 7 q^{2} + 36 q^{3} + 41 q^{4} + 101 q^{6} + 102 q^{7} + 15 q^{8} + 249 q^{9} + 363 q^{11} + 1237 q^{12} + 1646 q^{13} - 963 q^{14} - 2687 q^{16} + 1742 q^{17} + 3076 q^{18} - 10 q^{19} + 1236 q^{21}+ \cdots + 30129 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.45512 1.49467 0.747334 − 0.664448i \(-0.231333\pi\)
0.747334 + 0.664448i \(0.231333\pi\)
\(3\) 26.7755 1.71765 0.858824 − 0.512271i \(-0.171196\pi\)
0.858824 + 0.512271i \(0.171196\pi\)
\(4\) 39.4891 1.23403
\(5\) 0 0
\(6\) 226.390 2.56731
\(7\) −22.2927 −0.171956 −0.0859782 − 0.996297i \(-0.527402\pi\)
−0.0859782 + 0.996297i \(0.527402\pi\)
\(8\) 63.3212 0.349804
\(9\) 473.926 1.95031
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 1057.34 2.11964
\(13\) 225.547 0.370151 0.185075 − 0.982724i \(-0.440747\pi\)
0.185075 + 0.982724i \(0.440747\pi\)
\(14\) −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 \(0.353340\pi\)
\(18\) 4007.11 2.91507
\(19\) 2525.94 1.60524 0.802620 − 0.596491i \(-0.203439\pi\)
0.802620 + 0.596491i \(0.203439\pi\)
\(20\) 0 0
\(21\) −596.899 −0.295360
\(22\) 1023.07 0.450660
\(23\) 337.423 0.133001 0.0665006 − 0.997786i \(-0.478817\pi\)
0.0665006 + 0.997786i \(0.478817\pi\)
\(24\) 1695.46 0.600839
\(25\) 0 0
\(26\) 1907.03 0.553253
\(27\) 6183.16 1.63230
\(28\) −880.320 −0.212200
\(29\) −7644.60 −1.68795 −0.843976 − 0.536381i \(-0.819791\pi\)
−0.843976 + 0.536381i \(0.819791\pi\)
\(30\) 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
\(33\) 3239.83 0.517890
\(34\) 8958.95 1.32911
\(35\) 0 0
\(36\) 18714.9 2.40675
\(37\) 5663.43 0.680104 0.340052 − 0.940407i \(-0.389555\pi\)
0.340052 + 0.940407i \(0.389555\pi\)
\(38\) 21357.2 2.39930
\(39\) 6039.13 0.635789
\(40\) 0 0
\(41\) −13317.4 −1.23725 −0.618627 − 0.785685i \(-0.712311\pi\)
−0.618627 + 0.785685i \(0.712311\pi\)
\(42\) −5046.85 −0.441466
\(43\) −9007.03 −0.742866 −0.371433 − 0.928460i \(-0.621133\pi\)
−0.371433 + 0.928460i \(0.621133\pi\)
\(44\) 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 \(0.314808\pi\)
\(48\) −19499.6 −1.22158
\(49\) −16310.0 −0.970431
\(50\) 0 0
\(51\) 28371.0 1.52739
\(52\) 8906.65 0.456779
\(53\) −21261.8 −1.03971 −0.519853 − 0.854256i \(-0.674013\pi\)
−0.519853 + 0.854256i \(0.674013\pi\)
\(54\) 52279.4 2.43975
\(55\) 0 0
\(56\) −1411.60 −0.0601509
\(57\) 67633.4 2.75724
\(58\) −64636.1 −2.52293
\(59\) −44329.0 −1.65790 −0.828948 − 0.559325i \(-0.811060\pi\)
−0.828948 + 0.559325i \(0.811060\pi\)
\(60\) 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
\(63\) −10565.1 −0.335369
\(64\) −45890.9 −1.40048
\(65\) 0 0
\(66\) 27393.2 0.774074
\(67\) 45840.6 1.24757 0.623783 − 0.781598i \(-0.285595\pi\)
0.623783 + 0.781598i \(0.285595\pi\)
\(68\) 41842.2 1.09734
\(69\) 9034.68 0.228449
\(70\) 0 0
\(71\) −31877.6 −0.750481 −0.375240 − 0.926928i \(-0.622440\pi\)
−0.375240 + 0.926928i \(0.622440\pi\)
\(72\) 30009.6 0.682227
\(73\) −49936.7 −1.09676 −0.548382 − 0.836228i \(-0.684756\pi\)
−0.548382 + 0.836228i \(0.684756\pi\)
\(74\) 47885.0 1.01653
\(75\) 0 0
\(76\) 99747.3 1.98092
\(77\) −2697.42 −0.0518468
\(78\) 51061.6 0.950294
\(79\) 48257.1 0.869949 0.434974 − 0.900443i \(-0.356757\pi\)
0.434974 + 0.900443i \(0.356757\pi\)
\(80\) 0 0
\(81\) 50393.1 0.853411
\(82\) −112600. −1.84928
\(83\) −66052.0 −1.05242 −0.526212 − 0.850353i \(-0.676388\pi\)
−0.526212 + 0.850353i \(0.676388\pi\)
\(84\) −23571.0 −0.364485
\(85\) 0 0
\(86\) −76155.6 −1.11034
\(87\) −204688. −2.89931
\(88\) 7661.87 0.105470
\(89\) −124047. −1.66002 −0.830009 − 0.557750i \(-0.811665\pi\)
−0.830009 + 0.557750i \(0.811665\pi\)
\(90\) 0 0
\(91\) −5028.06 −0.0636498
\(92\) 13324.5 0.164128
\(93\) 180863. 2.16842
\(94\) 140729. 1.64272
\(95\) 0 0
\(96\) −219126. −2.42670
\(97\) −73854.3 −0.796978 −0.398489 − 0.917173i \(-0.630465\pi\)
−0.398489 + 0.917173i \(0.630465\pi\)
\(98\) −137903. −1.45047
\(99\) 57345.1 0.588042
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 275.6.a.c.1.3 3
5.2 odd 4 275.6.b.c.199.6 6
5.3 odd 4 275.6.b.c.199.1 6
5.4 even 2 55.6.a.a.1.1 ✓ 3
15.14 odd 2 495.6.a.f.1.3 3
20.19 odd 2 880.6.a.l.1.3 3
55.54 odd 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 5.4 even 2
275.6.a.c.1.3 3 1.1 even 1 trivial
275.6.b.c.199.1 6 5.3 odd 4
275.6.b.c.199.6 6 5.2 odd 4
495.6.a.f.1.3 3 15.14 odd 2
605.6.a.b.1.3 3 55.54 odd 2
880.6.a.l.1.3 3 20.19 odd 2