Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1386,2,Mod(1,1386)] 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("1386.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1386, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,0,2,0,0,-2,2,0,0,-2,0,4] 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: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.16228\) of defining polynomial
Character \(\chi\) \(=\) 1386.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.16228 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.16228 q^{10} -1.00000 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +0.837722 q^{17} -1.16228 q^{19} +3.16228 q^{20} -1.00000 q^{22} +6.32456 q^{23} +5.00000 q^{25} +2.00000 q^{26} -1.00000 q^{28} +4.00000 q^{29} +2.83772 q^{31} +1.00000 q^{32} +0.837722 q^{34} -3.16228 q^{35} -4.32456 q^{37} -1.16228 q^{38} +3.16228 q^{40} -0.837722 q^{41} +6.32456 q^{43} -1.00000 q^{44} +6.32456 q^{46} -7.48683 q^{47} +1.00000 q^{49} +5.00000 q^{50} +2.00000 q^{52} -8.32456 q^{53} -3.16228 q^{55} -1.00000 q^{56} +4.00000 q^{58} -14.3246 q^{59} +10.0000 q^{61} +2.83772 q^{62} +1.00000 q^{64} +6.32456 q^{65} -8.32456 q^{67} +0.837722 q^{68} -3.16228 q^{70} -8.00000 q^{71} +13.4868 q^{73} -4.32456 q^{74} -1.16228 q^{76} +1.00000 q^{77} +8.32456 q^{79} +3.16228 q^{80} -0.837722 q^{82} +1.16228 q^{83} +2.64911 q^{85} +6.32456 q^{86} -1.00000 q^{88} +3.67544 q^{89} -2.00000 q^{91} +6.32456 q^{92} -7.48683 q^{94} -3.67544 q^{95} +14.6491 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} - 2 q^{11} + 4 q^{13} - 2 q^{14} + 2 q^{16} + 8 q^{17} + 4 q^{19} - 2 q^{22} + 10 q^{25} + 4 q^{26} - 2 q^{28} + 8 q^{29} + 12 q^{31} + 2 q^{32} + 8 q^{34} + 4 q^{37}+ \cdots + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.16228 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 3.16228 1.00000
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.837722 0.203178 0.101589 − 0.994826i \(-0.467607\pi\)
0.101589 + 0.994826i \(0.467607\pi\)
\(18\) 0 0
\(19\) −1.16228 −0.266645 −0.133322 − 0.991073i \(-0.542565\pi\)
−0.133322 + 0.991073i \(0.542565\pi\)
\(20\) 3.16228 0.707107
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 6.32456 1.31876 0.659380 − 0.751809i \(-0.270819\pi\)
0.659380 + 0.751809i \(0.270819\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 2.83772 0.509670 0.254835 − 0.966985i \(-0.417979\pi\)
0.254835 + 0.966985i \(0.417979\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0.837722 0.143668
\(35\) −3.16228 −0.534522
\(36\) 0 0
\(37\) −4.32456 −0.710953 −0.355476 − 0.934685i \(-0.615681\pi\)
−0.355476 + 0.934685i \(0.615681\pi\)
\(38\) −1.16228 −0.188546
\(39\) 0 0
\(40\) 3.16228 0.500000
\(41\) −0.837722 −0.130830 −0.0654151 − 0.997858i \(-0.520837\pi\)
−0.0654151 + 0.997858i \(0.520837\pi\)
\(42\) 0 0
\(43\) 6.32456 0.964486 0.482243 − 0.876038i \(-0.339822\pi\)
0.482243 + 0.876038i \(0.339822\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 6.32456 0.932505
\(47\) −7.48683 −1.09207 −0.546033 − 0.837763i \(-0.683863\pi\)
−0.546033 + 0.837763i \(0.683863\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 5.00000 0.707107
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −8.32456 −1.14347 −0.571733 − 0.820440i \(-0.693729\pi\)
−0.571733 + 0.820440i \(0.693729\pi\)
\(54\) 0 0
\(55\) −3.16228 −0.426401
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 4.00000 0.525226
\(59\) −14.3246 −1.86490 −0.932449 − 0.361301i \(-0.882333\pi\)
−0.932449 + 0.361301i \(0.882333\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 2.83772 0.360391
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.32456 0.784465
\(66\) 0 0
\(67\) −8.32456 −1.01701 −0.508503 − 0.861060i \(-0.669801\pi\)
−0.508503 + 0.861060i \(0.669801\pi\)
\(68\) 0.837722 0.101589
\(69\) 0 0
\(70\) −3.16228 −0.377964
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 13.4868 1.57851 0.789257 − 0.614063i \(-0.210466\pi\)
0.789257 + 0.614063i \(0.210466\pi\)
\(74\) −4.32456 −0.502719
\(75\) 0 0
\(76\) −1.16228 −0.133322
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 8.32456 0.936586 0.468293 − 0.883573i \(-0.344869\pi\)
0.468293 + 0.883573i \(0.344869\pi\)
\(80\) 3.16228 0.353553
\(81\) 0 0
\(82\) −0.837722 −0.0925110
\(83\) 1.16228 0.127577 0.0637883 − 0.997963i \(-0.479682\pi\)
0.0637883 + 0.997963i \(0.479682\pi\)
\(84\) 0 0
\(85\) 2.64911 0.287336
\(86\) 6.32456 0.681994
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 3.67544 0.389596 0.194798 − 0.980843i \(-0.437595\pi\)
0.194798 + 0.980843i \(0.437595\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 6.32456 0.659380
\(93\) 0 0
\(94\) −7.48683 −0.772208
\(95\) −3.67544 −0.377093
\(96\) 0 0
\(97\) 14.6491 1.48739 0.743696 − 0.668518i \(-0.233071\pi\)
0.743696 + 0.668518i \(0.233071\pi\)
\(98\) 1.00000 0.101015
\(99\) 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 1386.2.a.o.1.2 yes 2
3.2 odd 2 1386.2.a.n.1.1 ✓ 2
7.6 odd 2 9702.2.a.dc.1.1 2
21.20 even 2 9702.2.a.cn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1386.2.a.n.1.1 ✓ 2 3.2 odd 2
1386.2.a.o.1.2 yes 2 1.1 even 1 trivial
9702.2.a.cn.1.2 2 21.20 even 2
9702.2.a.dc.1.1 2 7.6 odd 2