Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1323,4,Mod(1,1323)] 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("1323.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1323, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,44,0,0,0,0,0,-132,0,0,-336] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(78.0595269376\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 54x^{6} + 887x^{4} - 4176x^{2} + 3136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(4.67291\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.67291 q^{2} +13.8361 q^{4} -15.5408 q^{5} +27.2713 q^{8} -72.6208 q^{10} +63.4897 q^{11} -53.1931 q^{13} +16.7479 q^{16} -68.4660 q^{17} +86.0311 q^{19} -215.024 q^{20} +296.682 q^{22} +46.9229 q^{23} +116.517 q^{25} -248.567 q^{26} -169.768 q^{29} +141.842 q^{31} -139.909 q^{32} -319.935 q^{34} -411.923 q^{37} +402.015 q^{38} -423.819 q^{40} -49.0000 q^{41} -356.204 q^{43} +878.447 q^{44} +219.266 q^{46} -387.186 q^{47} +544.475 q^{50} -735.983 q^{52} -184.468 q^{53} -986.683 q^{55} -793.308 q^{58} -627.676 q^{59} -821.903 q^{61} +662.816 q^{62} -787.766 q^{64} +826.666 q^{65} -95.9610 q^{67} -947.298 q^{68} +733.093 q^{71} +750.676 q^{73} -1924.88 q^{74} +1190.33 q^{76} +23.5747 q^{79} -260.276 q^{80} -228.972 q^{82} -592.109 q^{83} +1064.02 q^{85} -1664.51 q^{86} +1731.45 q^{88} +864.292 q^{89} +649.227 q^{92} -1809.29 q^{94} -1337.00 q^{95} -614.268 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 44 q^{4} - 132 q^{10} - 336 q^{13} + 204 q^{16} + 288 q^{19} + 484 q^{22} + 152 q^{25} - 120 q^{31} - 1008 q^{34} + 592 q^{37} - 1620 q^{40} - 1872 q^{43} - 1644 q^{46} - 2400 q^{52} - 1344 q^{55}+ \cdots - 5712 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.67291 1.65212 0.826061 − 0.563581i \(-0.190577\pi\)
0.826061 + 0.563581i \(0.190577\pi\)
\(3\) 0 0
\(4\) 13.8361 1.72951
\(5\) −15.5408 −1.39001 −0.695007 − 0.719003i \(-0.744599\pi\)
−0.695007 + 0.719003i \(0.744599\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 27.2713 1.20523
\(9\) 0 0
\(10\) −72.6208 −2.29647
\(11\) 63.4897 1.74026 0.870131 − 0.492821i \(-0.164034\pi\)
0.870131 + 0.492821i \(0.164034\pi\)
\(12\) 0 0
\(13\) −53.1931 −1.13486 −0.567428 − 0.823423i \(-0.692061\pi\)
−0.567428 + 0.823423i \(0.692061\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.7479 0.261686
\(17\) −68.4660 −0.976790 −0.488395 − 0.872623i \(-0.662418\pi\)
−0.488395 + 0.872623i \(0.662418\pi\)
\(18\) 0 0
\(19\) 86.0311 1.03878 0.519392 − 0.854536i \(-0.326158\pi\)
0.519392 + 0.854536i \(0.326158\pi\)
\(20\) −215.024 −2.40404
\(21\) 0 0
\(22\) 296.682 2.87512
\(23\) 46.9229 0.425395 0.212698 − 0.977118i \(-0.431775\pi\)
0.212698 + 0.977118i \(0.431775\pi\)
\(24\) 0 0
\(25\) 116.517 0.932140
\(26\) −248.567 −1.87492
\(27\) 0 0
\(28\) 0 0
\(29\) −169.768 −1.08707 −0.543535 − 0.839386i \(-0.682915\pi\)
−0.543535 + 0.839386i \(0.682915\pi\)
\(30\) 0 0
\(31\) 141.842 0.821794 0.410897 − 0.911682i \(-0.365216\pi\)
0.410897 + 0.911682i \(0.365216\pi\)
\(32\) −139.909 −0.772897
\(33\) 0 0
\(34\) −319.935 −1.61378
\(35\) 0 0
\(36\) 0 0
\(37\) −411.923 −1.83026 −0.915132 − 0.403155i \(-0.867914\pi\)
−0.915132 + 0.403155i \(0.867914\pi\)
\(38\) 402.015 1.71620
\(39\) 0 0
\(40\) −423.819 −1.67529
\(41\) −49.0000 −0.186647 −0.0933234 − 0.995636i \(-0.529749\pi\)
−0.0933234 + 0.995636i \(0.529749\pi\)
\(42\) 0 0
\(43\) −356.204 −1.26327 −0.631636 − 0.775265i \(-0.717616\pi\)
−0.631636 + 0.775265i \(0.717616\pi\)
\(44\) 878.447 3.00979
\(45\) 0 0
\(46\) 219.266 0.702805
\(47\) −387.186 −1.20164 −0.600819 − 0.799385i \(-0.705159\pi\)
−0.600819 + 0.799385i \(0.705159\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 544.475 1.54001
\(51\) 0 0
\(52\) −735.983 −1.96274
\(53\) −184.468 −0.478088 −0.239044 − 0.971009i \(-0.576834\pi\)
−0.239044 + 0.971009i \(0.576834\pi\)
\(54\) 0 0
\(55\) −986.683 −2.41899
\(56\) 0 0
\(57\) 0 0
\(58\) −793.308 −1.79597
\(59\) −627.676 −1.38503 −0.692513 − 0.721406i \(-0.743496\pi\)
−0.692513 + 0.721406i \(0.743496\pi\)
\(60\) 0 0
\(61\) −821.903 −1.72515 −0.862573 − 0.505933i \(-0.831148\pi\)
−0.862573 + 0.505933i \(0.831148\pi\)
\(62\) 662.816 1.35770
\(63\) 0 0
\(64\) −787.766 −1.53860
\(65\) 826.666 1.57747
\(66\) 0 0
\(67\) −95.9610 −0.174978 −0.0874888 − 0.996166i \(-0.527884\pi\)
−0.0874888 + 0.996166i \(0.527884\pi\)
\(68\) −947.298 −1.68936
\(69\) 0 0
\(70\) 0 0
\(71\) 733.093 1.22538 0.612691 − 0.790322i \(-0.290087\pi\)
0.612691 + 0.790322i \(0.290087\pi\)
\(72\) 0 0
\(73\) 750.676 1.20356 0.601781 − 0.798661i \(-0.294458\pi\)
0.601781 + 0.798661i \(0.294458\pi\)
\(74\) −1924.88 −3.02382
\(75\) 0 0
\(76\) 1190.33 1.79658
\(77\) 0 0
\(78\) 0 0
\(79\) 23.5747 0.0335742 0.0167871 − 0.999859i \(-0.494656\pi\)
0.0167871 + 0.999859i \(0.494656\pi\)
\(80\) −260.276 −0.363747
\(81\) 0 0
\(82\) −228.972 −0.308363
\(83\) −592.109 −0.783042 −0.391521 − 0.920169i \(-0.628051\pi\)
−0.391521 + 0.920169i \(0.628051\pi\)
\(84\) 0 0
\(85\) 1064.02 1.35775
\(86\) −1664.51 −2.08708
\(87\) 0 0
\(88\) 1731.45 2.09742
\(89\) 864.292 1.02938 0.514690 − 0.857376i \(-0.327907\pi\)
0.514690 + 0.857376i \(0.327907\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 649.227 0.735724
\(93\) 0 0
\(94\) −1809.29 −1.98525
\(95\) −1337.00 −1.44392
\(96\) 0 0
\(97\) −614.268 −0.642984 −0.321492 − 0.946912i \(-0.604184\pi\)
−0.321492 + 0.946912i \(0.604184\pi\)
\(98\) 0 0
\(99\) 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 1323.4.a.bl.1.7 yes 8
3.2 odd 2 inner 1323.4.a.bl.1.2 ✓ 8
7.6 odd 2 1323.4.a.bm.1.7 yes 8
21.20 even 2 1323.4.a.bm.1.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1323.4.a.bl.1.2 ✓ 8 3.2 odd 2 inner
1323.4.a.bl.1.7 yes 8 1.1 even 1 trivial
1323.4.a.bm.1.2 yes 8 21.20 even 2
1323.4.a.bm.1.7 yes 8 7.6 odd 2