Properties

Label 3.18.a.b.1.2
Level $3$
Weight $18$
Character 3.1
Self dual yes
Analytic conductor $5.497$
Analytic rank $0$
Dimension $2$
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] = [3,18,Mod(1,3)] 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("3.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] 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: yes
Analytic conductor: \(5.49666262034\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14569}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3642 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-59.8511\) of defining polynomial
Character \(\chi\) \(=\) 3.1

$q$-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)
 
\(f(q)\) \(=\) \(q+659.106 q^{2} +6561.00 q^{3} +303349. q^{4} -1.08318e6 q^{5} +4.32440e6 q^{6} +1.59855e6 q^{7} +1.13549e8 q^{8} +4.30467e7 q^{9} -7.13934e8 q^{10} -4.47381e8 q^{11} +1.99027e9 q^{12} -2.48486e9 q^{13} +1.05361e9 q^{14} -7.10677e9 q^{15} +3.50803e10 q^{16} -2.48745e10 q^{17} +2.83724e10 q^{18} +8.23042e10 q^{19} -3.28583e11 q^{20} +1.04881e10 q^{21} -2.94872e11 q^{22} +6.43083e11 q^{23} +7.44995e11 q^{24} +4.10349e11 q^{25} -1.63779e12 q^{26} +2.82430e11 q^{27} +4.84918e11 q^{28} -9.82213e11 q^{29} -4.68412e12 q^{30} +3.28632e12 q^{31} +8.23853e12 q^{32} -2.93527e12 q^{33} -1.63949e13 q^{34} -1.73152e12 q^{35} +1.30582e13 q^{36} +2.63492e13 q^{37} +5.42472e13 q^{38} -1.63032e13 q^{39} -1.22994e14 q^{40} -3.33007e13 q^{41} +6.91276e12 q^{42} +9.83107e13 q^{43} -1.35713e14 q^{44} -4.66275e13 q^{45} +4.23860e14 q^{46} -1.62068e14 q^{47} +2.30162e14 q^{48} -2.30075e14 q^{49} +2.70463e14 q^{50} -1.63201e14 q^{51} -7.53780e14 q^{52} -1.40921e14 q^{53} +1.86151e14 q^{54} +4.84596e14 q^{55} +1.81514e14 q^{56} +5.39998e14 q^{57} -6.47383e14 q^{58} -9.80930e13 q^{59} -2.15583e15 q^{60} +1.37376e15 q^{61} +2.16603e15 q^{62} +6.88123e13 q^{63} +8.32030e14 q^{64} +2.69156e15 q^{65} -1.93465e15 q^{66} -1.85816e15 q^{67} -7.54565e15 q^{68} +4.21927e15 q^{69} -1.14126e15 q^{70} -6.17500e15 q^{71} +4.88791e15 q^{72} -1.30214e16 q^{73} +1.73670e16 q^{74} +2.69230e15 q^{75} +2.49669e16 q^{76} -7.15160e14 q^{77} -1.07455e16 q^{78} +1.27538e16 q^{79} -3.79984e16 q^{80} +1.85302e15 q^{81} -2.19487e16 q^{82} -1.42886e16 q^{83} +3.18155e15 q^{84} +2.69436e16 q^{85} +6.47972e16 q^{86} -6.44430e15 q^{87} -5.07996e16 q^{88} -3.77818e16 q^{89} -3.07325e16 q^{90} -3.97217e15 q^{91} +1.95079e17 q^{92} +2.15615e16 q^{93} -1.06820e17 q^{94} -8.91506e16 q^{95} +5.40530e16 q^{96} +1.09404e17 q^{97} -1.51644e17 q^{98} -1.92583e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 594 q^{2} + 13122 q^{3} + 176516 q^{4} + 382860 q^{5} + 3897234 q^{6} + 24471568 q^{7} + 130340232 q^{8} + 86093442 q^{9} - 809382420 q^{10} - 987553512 q^{11} + 1158121476 q^{12} - 2519398244 q^{13}+ \cdots - 42\!\cdots\!52 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 659.106 1.82054 0.910271 − 0.414014i \(-0.135873\pi\)
0.910271 + 0.414014i \(0.135873\pi\)
\(3\) 6561.00 0.577350
\(4\) 303349. 2.31437
\(5\) −1.08318e6 −1.24010 −0.620051 − 0.784562i \(-0.712888\pi\)
−0.620051 + 0.784562i \(0.712888\pi\)
\(6\) 4.32440e6 1.05109
\(7\) 1.59855e6 0.104808 0.0524038 − 0.998626i \(-0.483312\pi\)
0.0524038 + 0.998626i \(0.483312\pi\)
\(8\) 1.13549e8 2.39287
\(9\) 4.30467e7 0.333333
\(10\) −7.13934e8 −2.25766
\(11\) −4.47381e8 −0.629274 −0.314637 − 0.949212i \(-0.601883\pi\)
−0.314637 + 0.949212i \(0.601883\pi\)
\(12\) 1.99027e9 1.33620
\(13\) −2.48486e9 −0.844857 −0.422428 − 0.906396i \(-0.638822\pi\)
−0.422428 + 0.906396i \(0.638822\pi\)
\(14\) 1.05361e9 0.190806
\(15\) −7.10677e9 −0.715973
\(16\) 3.50803e10 2.04194
\(17\) −2.48745e10 −0.864844 −0.432422 − 0.901671i \(-0.642341\pi\)
−0.432422 + 0.901671i \(0.642341\pi\)
\(18\) 2.83724e10 0.606847
\(19\) 8.23042e10 1.11177 0.555887 − 0.831258i \(-0.312379\pi\)
0.555887 + 0.831258i \(0.312379\pi\)
\(20\) −3.28583e11 −2.87005
\(21\) 1.04881e10 0.0605106
\(22\) −2.94872e11 −1.14562
\(23\) 6.43083e11 1.71230 0.856151 − 0.516726i \(-0.172850\pi\)
0.856151 + 0.516726i \(0.172850\pi\)
\(24\) 7.44995e11 1.38152
\(25\) 4.10349e11 0.537852
\(26\) −1.63779e12 −1.53810
\(27\) 2.82430e11 0.192450
\(28\) 4.84918e11 0.242563
\(29\) −9.82213e11 −0.364605 −0.182302 − 0.983243i \(-0.558355\pi\)
−0.182302 + 0.983243i \(0.558355\pi\)
\(30\) −4.68412e12 −1.30346
\(31\) 3.28632e12 0.692046 0.346023 − 0.938226i \(-0.387532\pi\)
0.346023 + 0.938226i \(0.387532\pi\)
\(32\) 8.23853e12 1.32457
\(33\) −2.93527e12 −0.363311
\(34\) −1.63949e13 −1.57448
\(35\) −1.73152e12 −0.129972
\(36\) 1.30582e13 0.771457
\(37\) 2.63492e13 1.23326 0.616628 − 0.787254i \(-0.288498\pi\)
0.616628 + 0.787254i \(0.288498\pi\)
\(38\) 5.42472e13 2.02403
\(39\) −1.63032e13 −0.487778
\(40\) −1.22994e14 −2.96740
\(41\) −3.33007e13 −0.651314 −0.325657 − 0.945488i \(-0.605585\pi\)
−0.325657 + 0.945488i \(0.605585\pi\)
\(42\) 6.91276e12 0.110162
\(43\) 9.83107e13 1.28268 0.641341 − 0.767256i \(-0.278378\pi\)
0.641341 + 0.767256i \(0.278378\pi\)
\(44\) −1.35713e14 −1.45637
\(45\) −4.66275e13 −0.413367
\(46\) 4.23860e14 3.11732
\(47\) −1.62068e14 −0.992811 −0.496406 − 0.868091i \(-0.665347\pi\)
−0.496406 + 0.868091i \(0.665347\pi\)
\(48\) 2.30162e14 1.17891
\(49\) −2.30075e14 −0.989015
\(50\) 2.70463e14 0.979182
\(51\) −1.63201e14 −0.499318
\(52\) −7.53780e14 −1.95531
\(53\) −1.40921e14 −0.310907 −0.155453 − 0.987843i \(-0.549684\pi\)
−0.155453 + 0.987843i \(0.549684\pi\)
\(54\) 1.86151e14 0.350363
\(55\) 4.84596e14 0.780363
\(56\) 1.81514e14 0.250790
\(57\) 5.39998e14 0.641882
\(58\) −6.47383e14 −0.663778
\(59\) −9.80930e13 −0.0869753 −0.0434876 − 0.999054i \(-0.513847\pi\)
−0.0434876 + 0.999054i \(0.513847\pi\)
\(60\) −2.15583e15 −1.65703
\(61\) 1.37376e15 0.917503 0.458751 − 0.888565i \(-0.348297\pi\)
0.458751 + 0.888565i \(0.348297\pi\)
\(62\) 2.16603e15 1.25990
\(63\) 6.88123e13 0.0349358
\(64\) 8.32030e14 0.369495
\(65\) 2.69156e15 1.04771
\(66\) −1.93465e15 −0.661423
\(67\) −1.85816e15 −0.559046 −0.279523 − 0.960139i \(-0.590176\pi\)
−0.279523 + 0.960139i \(0.590176\pi\)
\(68\) −7.54565e15 −2.00157
\(69\) 4.21927e15 0.988598
\(70\) −1.14126e15 −0.236619
\(71\) −6.17500e15 −1.13486 −0.567428 − 0.823423i \(-0.692061\pi\)
−0.567428 + 0.823423i \(0.692061\pi\)
\(72\) 4.88791e15 0.797622
\(73\) −1.30214e16 −1.88979 −0.944896 − 0.327371i \(-0.893837\pi\)
−0.944896 + 0.327371i \(0.893837\pi\)
\(74\) 1.73670e16 2.24519
\(75\) 2.69230e15 0.310529
\(76\) 2.49669e16 2.57305
\(77\) −7.15160e14 −0.0659526
\(78\) −1.07455e16 −0.888021
\(79\) 1.27538e16 0.945824 0.472912 − 0.881110i \(-0.343203\pi\)
0.472912 + 0.881110i \(0.343203\pi\)
\(80\) −3.79984e16 −2.53221
\(81\) 1.85302e15 0.111111
\(82\) −2.19487e16 −1.18574
\(83\) −1.42886e16 −0.696348 −0.348174 − 0.937430i \(-0.613198\pi\)
−0.348174 + 0.937430i \(0.613198\pi\)
\(84\) 3.18155e15 0.140044
\(85\) 2.69436e16 1.07250
\(86\) 6.47972e16 2.33517
\(87\) −6.44430e15 −0.210505
\(88\) −5.07996e16 −1.50577
\(89\) −3.77818e16 −1.01734 −0.508672 − 0.860961i \(-0.669863\pi\)
−0.508672 + 0.860961i \(0.669863\pi\)
\(90\) −3.07325e16 −0.752552
\(91\) −3.97217e15 −0.0885473
\(92\) 1.95079e17 3.96290
\(93\) 2.15615e16 0.399553
\(94\) −1.06820e17 −1.80745
\(95\) −8.91506e16 −1.37871
\(96\) 5.40530e16 0.764741
\(97\) 1.09404e17 1.41733 0.708667 − 0.705543i \(-0.249297\pi\)
0.708667 + 0.705543i \(0.249297\pi\)
\(98\) −1.51644e17 −1.80054
\(99\) −1.92583e16 −0.209758
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 3.18.a.b.1.2 ✓ 2
3.2 odd 2 9.18.a.c.1.1 2
4.3 odd 2 48.18.a.h.1.1 2
5.2 odd 4 75.18.b.c.49.4 4
5.3 odd 4 75.18.b.c.49.1 4
5.4 even 2 75.18.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.18.a.b.1.2 ✓ 2 1.1 even 1 trivial
9.18.a.c.1.1 2 3.2 odd 2
48.18.a.h.1.1 2 4.3 odd 2
75.18.a.b.1.1 2 5.4 even 2
75.18.b.c.49.1 4 5.3 odd 4
75.18.b.c.49.4 4 5.2 odd 4