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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1350,2,Mod(143,1350)] 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("1350.143"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1350, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([2, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.q (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,-24,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.7798042729\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 450)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 557.1
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1350.557
Dual form 1350.2.q.c.143.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.965926 - 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-0.707107 - 0.707107i) q^{8} +(-3.00000 + 1.73205i) q^{11} +(-0.896575 - 3.34607i) q^{13} +(0.500000 + 0.866025i) q^{16} +(4.24264 - 4.24264i) q^{17} +5.00000i q^{19} +(3.34607 - 0.896575i) q^{22} +(-5.79555 + 1.55291i) q^{23} +3.46410i q^{26} +(-3.46410 - 6.00000i) q^{29} +(-2.00000 + 3.46410i) q^{31} +(-0.258819 - 0.965926i) q^{32} +(-5.19615 + 3.00000i) q^{34} +(-4.89898 - 4.89898i) q^{37} +(1.29410 - 4.82963i) q^{38} +(1.50000 + 0.866025i) q^{41} +(-11.7112 - 3.13801i) q^{43} -3.46410 q^{44} +6.00000 q^{46} +(5.79555 + 1.55291i) q^{47} +(6.06218 + 3.50000i) q^{49} +(0.896575 - 3.34607i) q^{52} +(-4.24264 - 4.24264i) q^{53} +(1.79315 + 6.69213i) q^{58} +(0.866025 - 1.50000i) q^{59} +(-4.00000 - 6.92820i) q^{61} +(2.82843 - 2.82843i) q^{62} +1.00000i q^{64} +(8.36516 - 2.24144i) q^{67} +(5.79555 - 1.55291i) q^{68} -6.92820i q^{71} +(-8.57321 + 8.57321i) q^{73} +(3.46410 + 6.00000i) q^{74} +(-2.50000 + 4.33013i) q^{76} +(-12.1244 + 7.00000i) q^{79} +(-1.22474 - 1.22474i) q^{82} +(2.32937 - 8.69333i) q^{83} +(10.5000 + 6.06218i) q^{86} +(3.34607 + 0.896575i) q^{88} -12.1244 q^{89} +(-5.79555 - 1.55291i) q^{92} +(-5.19615 - 3.00000i) q^{94} +(-1.34486 + 5.01910i) q^{97} +(-4.94975 - 4.94975i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{11} + 4 q^{16} - 16 q^{31} + 12 q^{41} + 48 q^{46} - 32 q^{61} - 20 q^{76} + 84 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.965926 − 0.258819i −0.683013 − 0.183013i
\(3\) 0 0
\(4\) 0.866025 + 0.500000i 0.433013 + 0.250000i
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 −0.965926 − 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(8\) −0.707107 − 0.707107i −0.250000 − 0.250000i
\(9\) 0 0
\(10\) 0 0
\(11\) −3.00000 + 1.73205i −0.904534 + 0.522233i −0.878668 − 0.477432i \(-0.841568\pi\)
−0.0258656 + 0.999665i \(0.508234\pi\)
\(12\) 0 0
\(13\) −0.896575 − 3.34607i −0.248665 − 0.928032i −0.971506 − 0.237016i \(-0.923830\pi\)
0.722840 − 0.691015i \(-0.242836\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.500000 + 0.866025i 0.125000 + 0.216506i
\(17\) 4.24264 − 4.24264i 1.02899 − 1.02899i 0.0294245 − 0.999567i \(-0.490633\pi\)
0.999567 − 0.0294245i \(-0.00936746\pi\)
\(18\) 0 0
\(19\) 5.00000i 1.14708i 0.819178 + 0.573539i \(0.194430\pi\)
−0.819178 + 0.573539i \(0.805570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.34607 − 0.896575i 0.713384 − 0.191151i
\(23\) −5.79555 + 1.55291i −1.20846 + 0.323805i −0.806156 − 0.591703i \(-0.798456\pi\)
−0.402300 + 0.915508i \(0.631789\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3.46410i 0.679366i
\(27\) 0 0
\(28\) 0 0
\(29\) −3.46410 − 6.00000i −0.643268 − 1.11417i −0.984699 − 0.174265i \(-0.944245\pi\)
0.341431 − 0.939907i \(-0.389088\pi\)
\(30\) 0 0
\(31\) −2.00000 + 3.46410i −0.359211 + 0.622171i −0.987829 − 0.155543i \(-0.950287\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) −0.258819 − 0.965926i −0.0457532 − 0.170753i
\(33\) 0 0
\(34\) −5.19615 + 3.00000i −0.891133 + 0.514496i
\(35\) 0 0
\(36\) 0 0
\(37\) −4.89898 − 4.89898i −0.805387 − 0.805387i 0.178545 − 0.983932i \(-0.442861\pi\)
−0.983932 + 0.178545i \(0.942861\pi\)
\(38\) 1.29410 − 4.82963i 0.209930 − 0.783469i
\(39\) 0 0
\(40\) 0 0
\(41\) 1.50000 + 0.866025i 0.234261 + 0.135250i 0.612536 − 0.790443i \(-0.290149\pi\)
−0.378275 + 0.925693i \(0.623483\pi\)
\(42\) 0 0
\(43\) −11.7112 − 3.13801i −1.78595 − 0.478543i −0.794299 − 0.607527i \(-0.792162\pi\)
−0.991647 + 0.128984i \(0.958828\pi\)
\(44\) −3.46410 −0.522233
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 5.79555 + 1.55291i 0.845369 + 0.226516i 0.655407 − 0.755276i \(-0.272497\pi\)
0.189961 + 0.981792i \(0.439164\pi\)
\(48\) 0 0
\(49\) 6.06218 + 3.50000i 0.866025 + 0.500000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0.896575 − 3.34607i 0.124333 − 0.464016i
\(53\) −4.24264 − 4.24264i −0.582772 − 0.582772i 0.352892 − 0.935664i \(-0.385198\pi\)
−0.935664 + 0.352892i \(0.885198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 1.79315 + 6.69213i 0.235452 + 0.878720i
\(59\) 0.866025 − 1.50000i 0.112747 − 0.195283i −0.804130 − 0.594454i \(-0.797368\pi\)
0.916877 + 0.399170i \(0.130702\pi\)
\(60\) 0 0
\(61\) −4.00000 − 6.92820i −0.512148 − 0.887066i −0.999901 − 0.0140840i \(-0.995517\pi\)
0.487753 − 0.872982i \(-0.337817\pi\)
\(62\) 2.82843 − 2.82843i 0.359211 − 0.359211i
\(63\) 0 0
\(64\) 1.00000i 0.125000i
\(65\) 0 0
\(66\) 0 0
\(67\) 8.36516 − 2.24144i 1.02197 − 0.273835i 0.291346 − 0.956618i \(-0.405897\pi\)
0.730622 + 0.682783i \(0.239230\pi\)
\(68\) 5.79555 − 1.55291i 0.702814 − 0.188319i
\(69\) 0 0
\(70\) 0 0
\(71\) − 6.92820i − 0.822226i −0.911584 − 0.411113i \(-0.865140\pi\)
0.911584 − 0.411113i \(-0.134860\pi\)
\(72\) 0 0
\(73\) −8.57321 + 8.57321i −1.00342 + 1.00342i −0.00342468 + 0.999994i \(0.501090\pi\)
−0.999994 + 0.00342468i \(0.998910\pi\)
\(74\) 3.46410 + 6.00000i 0.402694 + 0.697486i
\(75\) 0 0
\(76\) −2.50000 + 4.33013i −0.286770 + 0.496700i
\(77\) 0 0
\(78\) 0 0
\(79\) −12.1244 + 7.00000i −1.36410 + 0.787562i −0.990166 − 0.139895i \(-0.955323\pi\)
−0.373930 + 0.927457i \(0.621990\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.22474 − 1.22474i −0.135250 − 0.135250i
\(83\) 2.32937 − 8.69333i 0.255682 − 0.954217i −0.712028 − 0.702151i \(-0.752223\pi\)
0.967710 − 0.252066i \(-0.0811101\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.5000 + 6.06218i 1.13224 + 0.653701i
\(87\) 0 0
\(88\) 3.34607 + 0.896575i 0.356692 + 0.0955753i
\(89\) −12.1244 −1.28518 −0.642590 − 0.766211i \(-0.722140\pi\)
−0.642590 + 0.766211i \(0.722140\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −5.79555 − 1.55291i −0.604228 − 0.161903i
\(93\) 0 0
\(94\) −5.19615 − 3.00000i −0.535942 − 0.309426i
\(95\) 0 0
\(96\) 0 0
\(97\) −1.34486 + 5.01910i −0.136550 + 0.509612i 0.863437 + 0.504457i \(0.168307\pi\)
−0.999987 + 0.00515471i \(0.998359\pi\)
\(98\) −4.94975 − 4.94975i −0.500000 − 0.500000i
\(99\) 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 1350.2.q.c.557.1 8
3.2 odd 2 450.2.p.g.257.2 yes 8
5.2 odd 4 inner 1350.2.q.c.1043.2 8
5.3 odd 4 inner 1350.2.q.c.1043.1 8
5.4 even 2 inner 1350.2.q.c.557.2 8
9.2 odd 6 inner 1350.2.q.c.1007.1 8
9.7 even 3 450.2.p.g.407.2 yes 8
15.2 even 4 450.2.p.g.293.1 yes 8
15.8 even 4 450.2.p.g.293.2 yes 8
15.14 odd 2 450.2.p.g.257.1 ✓ 8
45.2 even 12 inner 1350.2.q.c.143.2 8
45.7 odd 12 450.2.p.g.443.1 yes 8
45.29 odd 6 inner 1350.2.q.c.1007.2 8
45.34 even 6 450.2.p.g.407.1 yes 8
45.38 even 12 inner 1350.2.q.c.143.1 8
45.43 odd 12 450.2.p.g.443.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.2.p.g.257.1 ✓ 8 15.14 odd 2
450.2.p.g.257.2 yes 8 3.2 odd 2
450.2.p.g.293.1 yes 8 15.2 even 4
450.2.p.g.293.2 yes 8 15.8 even 4
450.2.p.g.407.1 yes 8 45.34 even 6
450.2.p.g.407.2 yes 8 9.7 even 3
450.2.p.g.443.1 yes 8 45.7 odd 12
450.2.p.g.443.2 yes 8 45.43 odd 12
1350.2.q.c.143.1 8 45.38 even 12 inner
1350.2.q.c.143.2 8 45.2 even 12 inner
1350.2.q.c.557.1 8 1.1 even 1 trivial
1350.2.q.c.557.2 8 5.4 even 2 inner
1350.2.q.c.1007.1 8 9.2 odd 6 inner
1350.2.q.c.1007.2 8 45.29 odd 6 inner
1350.2.q.c.1043.1 8 5.3 odd 4 inner
1350.2.q.c.1043.2 8 5.2 odd 4 inner