Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,4,Mod(11,72)] 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("72.11"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 72.l (of order \(6\), degree \(2\), 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: \(4.24813752041\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 59.2
Root \(1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 72.59
Dual form 72.4.l.a.11.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.44949 - 1.41421i) q^{2} +(-1.27526 - 5.03723i) q^{3} +(4.00000 - 6.92820i) q^{4} +(-10.2474 - 10.5352i) q^{6} -22.6274i q^{8} +(-23.7474 + 12.8475i) q^{9} +(17.1186 - 9.88344i) q^{11} +(-40.0000 - 11.3137i) q^{12} +(-32.0000 - 55.4256i) q^{16} +24.2022i q^{17} +(-40.0000 + 65.0538i) q^{18} +163.227 q^{19} +(27.9546 - 48.4188i) q^{22} +(-113.980 + 28.8557i) q^{24} +(62.5000 + 108.253i) q^{25} +(95.0000 + 103.238i) q^{27} +(-156.767 - 90.5097i) q^{32} +(-71.6158 - 73.6266i) q^{33} +(34.2270 + 59.2830i) q^{34} +(-5.97959 + 215.917i) q^{36} +(399.823 - 230.838i) q^{38} +(-367.005 - 211.890i) q^{41} +(281.931 + 488.319i) q^{43} -158.135i q^{44} +(-238.384 + 231.873i) q^{48} +(-171.500 + 297.047i) q^{49} +(306.186 + 176.777i) q^{50} +(121.912 - 30.8639i) q^{51} +(378.702 + 118.529i) q^{54} +(-208.156 - 822.213i) q^{57} +(-775.346 - 447.646i) q^{59} -512.000 q^{64} +(-279.546 - 79.0675i) q^{66} +(456.476 - 790.640i) q^{67} +(167.678 + 96.8087i) q^{68} +(290.706 + 537.343i) q^{72} -799.089 q^{73} +(465.593 - 452.878i) q^{75} +(652.908 - 1130.87i) q^{76} +(398.883 - 610.191i) q^{81} -1198.63 q^{82} +(-590.327 + 340.825i) q^{83} +(1381.18 + 797.422i) q^{86} +(-223.637 - 387.350i) q^{88} +1329.36i q^{89} +(-256.000 + 905.097i) q^{96} +(455.455 + 788.870i) q^{97} +970.151i q^{98} +(-279.546 + 454.638i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{3} + 16 q^{4} + 8 q^{6} - 46 q^{9} - 54 q^{11} - 160 q^{12} - 128 q^{16} - 160 q^{18} + 212 q^{19} + 200 q^{22} - 64 q^{24} + 250 q^{25} + 380 q^{27} + 370 q^{33} - 304 q^{34} + 368 q^{36} + 1080 q^{38}+ \cdots - 2000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.44949 − 1.41421i 0.866025 − 0.500000i
\(3\) −1.27526 − 5.03723i −0.245423 − 0.969416i
\(4\) 4.00000 − 6.92820i 0.500000 − 0.866025i
\(5\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) −10.2474 − 10.5352i −0.697251 − 0.716827i
\(7\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) − 22.6274i − 1.00000i
\(9\) −23.7474 + 12.8475i −0.879535 + 0.475834i
\(10\) 0 0
\(11\) 17.1186 − 9.88344i 0.469224 − 0.270906i −0.246691 − 0.969094i \(-0.579343\pi\)
0.715915 + 0.698188i \(0.246010\pi\)
\(12\) −40.0000 − 11.3137i −0.962250 − 0.272166i
\(13\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −32.0000 − 55.4256i −0.500000 − 0.866025i
\(17\) 24.2022i 0.345288i 0.984984 + 0.172644i \(0.0552309\pi\)
−0.984984 + 0.172644i \(0.944769\pi\)
\(18\) −40.0000 + 65.0538i −0.523783 + 0.851852i
\(19\) 163.227 1.97089 0.985443 − 0.170004i \(-0.0543779\pi\)
0.985443 + 0.170004i \(0.0543779\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 27.9546 − 48.4188i 0.270906 − 0.469224i
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) −113.980 + 28.8557i −0.969416 + 0.245423i
\(25\) 62.5000 + 108.253i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 95.0000 + 103.238i 0.677139 + 0.735855i
\(28\) 0 0
\(29\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) −156.767 − 90.5097i −0.866025 − 0.500000i
\(33\) −71.6158 − 73.6266i −0.377779 − 0.388386i
\(34\) 34.2270 + 59.2830i 0.172644 + 0.299028i
\(35\) 0 0
\(36\) −5.97959 + 215.917i −0.0276833 + 0.999617i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 399.823 − 230.838i 1.70684 − 0.985443i
\(39\) 0 0
\(40\) 0 0
\(41\) −367.005 − 211.890i −1.39797 − 0.807116i −0.403786 − 0.914854i \(-0.632306\pi\)
−0.994179 + 0.107738i \(0.965639\pi\)
\(42\) 0 0
\(43\) 281.931 + 488.319i 0.999864 + 1.73181i 0.514239 + 0.857647i \(0.328074\pi\)
0.485624 + 0.874168i \(0.338592\pi\)
\(44\) − 158.135i − 0.541813i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) −238.384 + 231.873i −0.716827 + 0.697251i
\(49\) −171.500 + 297.047i −0.500000 + 0.866025i
\(50\) 306.186 + 176.777i 0.866025 + 0.500000i
\(51\) 121.912 − 30.8639i 0.334727 − 0.0847415i
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 378.702 + 118.529i 0.954347 + 0.298699i
\(55\) 0 0
\(56\) 0 0
\(57\) −208.156 − 822.213i −0.483701 − 1.91061i
\(58\) 0 0
\(59\) −775.346 − 447.646i −1.71087 − 0.987772i −0.933388 − 0.358868i \(-0.883163\pi\)
−0.777483 − 0.628904i \(-0.783504\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) 0 0
\(66\) −279.546 − 79.0675i −0.521359 − 0.147463i
\(67\) 456.476 − 790.640i 0.832350 − 1.44167i −0.0638199 − 0.997961i \(-0.520328\pi\)
0.896170 − 0.443711i \(-0.146338\pi\)
\(68\) 167.678 + 96.8087i 0.299028 + 0.172644i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 290.706 + 537.343i 0.475834 + 0.879535i
\(73\) −799.089 −1.28118 −0.640591 − 0.767882i \(-0.721311\pi\)
−0.640591 + 0.767882i \(0.721311\pi\)
\(74\) 0 0
\(75\) 465.593 − 452.878i 0.716827 − 0.697251i
\(76\) 652.908 − 1130.87i 0.985443 − 1.70684i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) 398.883 − 610.191i 0.547164 − 0.837025i
\(82\) −1198.63 −1.61423
\(83\) −590.327 + 340.825i −0.780684 + 0.450728i −0.836673 − 0.547703i \(-0.815502\pi\)
0.0559884 + 0.998431i \(0.482169\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1381.18 + 797.422i 1.73181 + 0.999864i
\(87\) 0 0
\(88\) −223.637 − 387.350i −0.270906 − 0.469224i
\(89\) 1329.36i 1.58328i 0.610988 + 0.791640i \(0.290773\pi\)
−0.610988 + 0.791640i \(0.709227\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −256.000 + 905.097i −0.272166 + 0.962250i
\(97\) 455.455 + 788.870i 0.476746 + 0.825749i 0.999645 − 0.0266459i \(-0.00848265\pi\)
−0.522898 + 0.852395i \(0.675149\pi\)
\(98\) 970.151i 1.00000i
\(99\) −279.546 + 454.638i −0.283792 + 0.461544i
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 72.4.l.a.59.2 yes 4
3.2 odd 2 216.4.l.a.179.1 4
4.3 odd 2 288.4.p.a.239.1 4
8.3 odd 2 CM 72.4.l.a.59.2 yes 4
8.5 even 2 288.4.p.a.239.1 4
9.2 odd 6 inner 72.4.l.a.11.2 ✓ 4
9.7 even 3 216.4.l.a.35.1 4
12.11 even 2 864.4.p.a.719.2 4
24.5 odd 2 864.4.p.a.719.2 4
24.11 even 2 216.4.l.a.179.1 4
36.7 odd 6 864.4.p.a.143.2 4
36.11 even 6 288.4.p.a.47.1 4
72.11 even 6 inner 72.4.l.a.11.2 ✓ 4
72.29 odd 6 288.4.p.a.47.1 4
72.43 odd 6 216.4.l.a.35.1 4
72.61 even 6 864.4.p.a.143.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.4.l.a.11.2 ✓ 4 9.2 odd 6 inner
72.4.l.a.11.2 ✓ 4 72.11 even 6 inner
72.4.l.a.59.2 yes 4 1.1 even 1 trivial
72.4.l.a.59.2 yes 4 8.3 odd 2 CM
216.4.l.a.35.1 4 9.7 even 3
216.4.l.a.35.1 4 72.43 odd 6
216.4.l.a.179.1 4 3.2 odd 2
216.4.l.a.179.1 4 24.11 even 2
288.4.p.a.47.1 4 36.11 even 6
288.4.p.a.47.1 4 72.29 odd 6
288.4.p.a.239.1 4 4.3 odd 2
288.4.p.a.239.1 4 8.5 even 2
864.4.p.a.143.2 4 36.7 odd 6
864.4.p.a.143.2 4 72.61 even 6
864.4.p.a.719.2 4 12.11 even 2
864.4.p.a.719.2 4 24.5 odd 2