Properties

Label 378.2.a.b.1.1
Level $378$
Weight $2$
Character 378.1
Self dual yes
Analytic conductor $3.018$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [378,2,Mod(1,378)] 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("378.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(378, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,0,1,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.01834519640\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 378.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{10} +3.00000 q^{11} -4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -7.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} -3.00000 q^{23} +4.00000 q^{25} +4.00000 q^{26} +1.00000 q^{28} +5.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} -3.00000 q^{35} -7.00000 q^{37} +7.00000 q^{38} +3.00000 q^{40} -9.00000 q^{41} -10.0000 q^{43} +3.00000 q^{44} +3.00000 q^{46} +6.00000 q^{47} +1.00000 q^{49} -4.00000 q^{50} -4.00000 q^{52} +12.0000 q^{53} -9.00000 q^{55} -1.00000 q^{56} -6.00000 q^{59} +8.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} +12.0000 q^{65} -4.00000 q^{67} -6.00000 q^{68} +3.00000 q^{70} +9.00000 q^{71} +2.00000 q^{73} +7.00000 q^{74} -7.00000 q^{76} +3.00000 q^{77} -10.0000 q^{79} -3.00000 q^{80} +9.00000 q^{82} +18.0000 q^{85} +10.0000 q^{86} -3.00000 q^{88} +15.0000 q^{89} -4.00000 q^{91} -3.00000 q^{92} -6.00000 q^{94} +21.0000 q^{95} +8.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.00000 0.948683
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 − 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 − 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 7.00000 1.13555
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 12.0000 1.64833 0.824163 − 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) −9.00000 −1.21356
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −5.00000 −0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 3.00000 0.358569
\(71\) 9.00000 1.06810 0.534052 − 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 7.00000 0.813733
\(75\) 0 0
\(76\) −7.00000 −0.802955
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −3.00000 −0.335410
\(81\) 0 0
\(82\) 9.00000 0.993884
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 18.0000 1.95237
\(86\) 10.0000 1.07833
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 15.0000 1.59000 0.794998 − 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) −3.00000 −0.312772
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 21.0000 2.15455
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −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 378.2.a.b.1.1 ✓ 1
3.2 odd 2 378.2.a.g.1.1 yes 1
4.3 odd 2 3024.2.a.c.1.1 1
5.4 even 2 9450.2.a.cu.1.1 1
7.6 odd 2 2646.2.a.n.1.1 1
9.2 odd 6 1134.2.f.b.757.1 2
9.4 even 3 1134.2.f.o.379.1 2
9.5 odd 6 1134.2.f.b.379.1 2
9.7 even 3 1134.2.f.o.757.1 2
12.11 even 2 3024.2.a.bb.1.1 1
15.14 odd 2 9450.2.a.h.1.1 1
21.20 even 2 2646.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.a.b.1.1 ✓ 1 1.1 even 1 trivial
378.2.a.g.1.1 yes 1 3.2 odd 2
1134.2.f.b.379.1 2 9.5 odd 6
1134.2.f.b.757.1 2 9.2 odd 6
1134.2.f.o.379.1 2 9.4 even 3
1134.2.f.o.757.1 2 9.7 even 3
2646.2.a.n.1.1 1 7.6 odd 2
2646.2.a.q.1.1 1 21.20 even 2
3024.2.a.c.1.1 1 4.3 odd 2
3024.2.a.bb.1.1 1 12.11 even 2
9450.2.a.h.1.1 1 15.14 odd 2
9450.2.a.cu.1.1 1 5.4 even 2