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Properties

Label 15.18.a.a.1.2
Level $15$
Weight $18$
Character 15.1
Self dual yes
Analytic conductor $27.483$
Analytic rank $1$
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] = [15,18,Mod(1,15)] 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("15.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 15.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-356] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(27.4833131017\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{849}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 212 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-14.0688\) of defining polynomial
Character \(\chi\) \(=\) 15.1

$q$-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)
 
\(f(q)\) \(=\) \(q+346.477 q^{2} +6561.00 q^{3} -11025.8 q^{4} +390625. q^{5} +2.27323e6 q^{6} -1.80797e7 q^{7} -4.92336e7 q^{8} +4.30467e7 q^{9} +1.35343e8 q^{10} -1.45296e8 q^{11} -7.23401e7 q^{12} -3.26155e9 q^{13} -6.26421e9 q^{14} +2.56289e9 q^{15} -1.56131e10 q^{16} +7.81194e9 q^{17} +1.49147e10 q^{18} -7.18160e10 q^{19} -4.30694e9 q^{20} -1.18621e11 q^{21} -5.03416e10 q^{22} +1.67169e11 q^{23} -3.23022e11 q^{24} +1.52588e11 q^{25} -1.13005e12 q^{26} +2.82430e11 q^{27} +1.99343e11 q^{28} +1.63211e11 q^{29} +8.87982e11 q^{30} -5.38766e12 q^{31} +1.04356e12 q^{32} -9.53285e11 q^{33} +2.70666e12 q^{34} -7.06240e12 q^{35} -4.74623e11 q^{36} -3.68759e13 q^{37} -2.48826e13 q^{38} -2.13991e13 q^{39} -1.92319e13 q^{40} -7.52018e12 q^{41} -4.10995e13 q^{42} -5.73935e13 q^{43} +1.60200e12 q^{44} +1.68151e13 q^{45} +5.79201e13 q^{46} +1.42516e14 q^{47} -1.02438e14 q^{48} +9.42466e13 q^{49} +5.28682e13 q^{50} +5.12542e13 q^{51} +3.59611e13 q^{52} +8.01583e14 q^{53} +9.78553e13 q^{54} -5.67561e13 q^{55} +8.90131e14 q^{56} -4.71185e14 q^{57} +5.65490e13 q^{58} +9.64437e14 q^{59} -2.82578e13 q^{60} -1.29792e15 q^{61} -1.86670e15 q^{62} -7.78274e14 q^{63} +2.40801e15 q^{64} -1.27404e15 q^{65} -3.30291e14 q^{66} +4.71635e15 q^{67} -8.61327e13 q^{68} +1.09679e15 q^{69} -2.44696e15 q^{70} +9.29884e15 q^{71} -2.11934e15 q^{72} -6.86407e15 q^{73} -1.27767e16 q^{74} +1.00113e15 q^{75} +7.91827e14 q^{76} +2.62691e15 q^{77} -7.41428e15 q^{78} +1.21446e16 q^{79} -6.09888e15 q^{80} +1.85302e15 q^{81} -2.60557e15 q^{82} -3.23986e16 q^{83} +1.30789e15 q^{84} +3.05154e15 q^{85} -1.98855e16 q^{86} +1.07083e15 q^{87} +7.15343e15 q^{88} -4.91787e16 q^{89} +5.82605e15 q^{90} +5.89681e16 q^{91} -1.84316e15 q^{92} -3.53484e16 q^{93} +4.93784e16 q^{94} -2.80531e16 q^{95} +6.84677e15 q^{96} +1.26124e17 q^{97} +3.26543e16 q^{98} -6.25450e15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 356 q^{2} + 13122 q^{3} + 351376 q^{4} + 781250 q^{5} - 2335716 q^{6} - 20754552 q^{7} - 211737408 q^{8} + 86093442 q^{9} - 139062500 q^{10} - 1131629912 q^{11} + 2305377936 q^{12} - 446672524 q^{13}+ \cdots - 48\!\cdots\!52 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 346.477 0.957016 0.478508 − 0.878083i \(-0.341178\pi\)
0.478508 + 0.878083i \(0.341178\pi\)
\(3\) 6561.00 0.577350
\(4\) −11025.8 −0.0841199
\(5\) 390625. 0.447214
\(6\) 2.27323e6 0.552534
\(7\) −1.80797e7 −1.18538 −0.592692 − 0.805429i \(-0.701935\pi\)
−0.592692 + 0.805429i \(0.701935\pi\)
\(8\) −4.92336e7 −1.03752
\(9\) 4.30467e7 0.333333
\(10\) 1.35343e8 0.427991
\(11\) −1.45296e8 −0.204369 −0.102184 − 0.994765i \(-0.532583\pi\)
−0.102184 + 0.994765i \(0.532583\pi\)
\(12\) −7.23401e7 −0.0485667
\(13\) −3.26155e9 −1.10894 −0.554468 − 0.832205i \(-0.687078\pi\)
−0.554468 + 0.832205i \(0.687078\pi\)
\(14\) −6.26421e9 −1.13443
\(15\) 2.56289e9 0.258199
\(16\) −1.56131e10 −0.908804
\(17\) 7.81194e9 0.271608 0.135804 − 0.990736i \(-0.456638\pi\)
0.135804 + 0.990736i \(0.456638\pi\)
\(18\) 1.49147e10 0.319005
\(19\) −7.18160e10 −0.970098 −0.485049 − 0.874487i \(-0.661198\pi\)
−0.485049 + 0.874487i \(0.661198\pi\)
\(20\) −4.30694e9 −0.0376196
\(21\) −1.18621e11 −0.684382
\(22\) −5.03416e10 −0.195584
\(23\) 1.67169e11 0.445111 0.222555 − 0.974920i \(-0.428560\pi\)
0.222555 + 0.974920i \(0.428560\pi\)
\(24\) −3.23022e11 −0.599013
\(25\) 1.52588e11 0.200000
\(26\) −1.13005e12 −1.06127
\(27\) 2.82430e11 0.192450
\(28\) 1.99343e11 0.0997144
\(29\) 1.63211e11 0.0605854 0.0302927 − 0.999541i \(-0.490356\pi\)
0.0302927 + 0.999541i \(0.490356\pi\)
\(30\) 8.87982e11 0.247101
\(31\) −5.38766e12 −1.13456 −0.567278 − 0.823527i \(-0.692003\pi\)
−0.567278 + 0.823527i \(0.692003\pi\)
\(32\) 1.04356e12 0.167780
\(33\) −9.53285e11 −0.117992
\(34\) 2.70666e12 0.259934
\(35\) −7.06240e12 −0.530120
\(36\) −4.74623e11 −0.0280400
\(37\) −3.68759e13 −1.72595 −0.862976 − 0.505246i \(-0.831402\pi\)
−0.862976 + 0.505246i \(0.831402\pi\)
\(38\) −2.48826e13 −0.928400
\(39\) −2.13991e13 −0.640244
\(40\) −1.92319e13 −0.463993
\(41\) −7.52018e12 −0.147084 −0.0735420 − 0.997292i \(-0.523430\pi\)
−0.0735420 + 0.997292i \(0.523430\pi\)
\(42\) −4.10995e13 −0.654964
\(43\) −5.73935e13 −0.748825 −0.374413 − 0.927262i \(-0.622156\pi\)
−0.374413 + 0.927262i \(0.622156\pi\)
\(44\) 1.60200e12 0.0171915
\(45\) 1.68151e13 0.149071
\(46\) 5.79201e13 0.425978
\(47\) 1.42516e14 0.873034 0.436517 − 0.899696i \(-0.356212\pi\)
0.436517 + 0.899696i \(0.356212\pi\)
\(48\) −1.02438e14 −0.524698
\(49\) 9.42466e13 0.405134
\(50\) 5.28682e13 0.191403
\(51\) 5.12542e13 0.156813
\(52\) 3.59611e13 0.0932836
\(53\) 8.01583e14 1.76849 0.884247 − 0.467020i \(-0.154672\pi\)
0.884247 + 0.467020i \(0.154672\pi\)
\(54\) 9.78553e13 0.184178
\(55\) −5.67561e13 −0.0913966
\(56\) 8.90131e14 1.22986
\(57\) −4.71185e14 −0.560087
\(58\) 5.65490e13 0.0579812
\(59\) 9.64437e14 0.855129 0.427565 − 0.903985i \(-0.359372\pi\)
0.427565 + 0.903985i \(0.359372\pi\)
\(60\) −2.82578e13 −0.0217197
\(61\) −1.29792e15 −0.866849 −0.433424 − 0.901190i \(-0.642695\pi\)
−0.433424 + 0.901190i \(0.642695\pi\)
\(62\) −1.86670e15 −1.08579
\(63\) −7.78274e14 −0.395128
\(64\) 2.40801e15 1.06937
\(65\) −1.27404e15 −0.495931
\(66\) −3.30291e14 −0.112921
\(67\) 4.71635e15 1.41896 0.709480 − 0.704726i \(-0.248930\pi\)
0.709480 + 0.704726i \(0.248930\pi\)
\(68\) −8.61327e13 −0.0228477
\(69\) 1.09679e15 0.256985
\(70\) −2.44696e15 −0.507333
\(71\) 9.29884e15 1.70896 0.854482 − 0.519482i \(-0.173875\pi\)
0.854482 + 0.519482i \(0.173875\pi\)
\(72\) −2.11934e15 −0.345840
\(73\) −6.86407e15 −0.996180 −0.498090 − 0.867125i \(-0.665965\pi\)
−0.498090 + 0.867125i \(0.665965\pi\)
\(74\) −1.27767e16 −1.65176
\(75\) 1.00113e15 0.115470
\(76\) 7.91827e14 0.0816046
\(77\) 2.62691e15 0.242256
\(78\) −7.41428e15 −0.612724
\(79\) 1.21446e16 0.900646 0.450323 − 0.892866i \(-0.351309\pi\)
0.450323 + 0.892866i \(0.351309\pi\)
\(80\) −6.09888e15 −0.406429
\(81\) 1.85302e15 0.111111
\(82\) −2.60557e15 −0.140762
\(83\) −3.23986e16 −1.57893 −0.789465 − 0.613796i \(-0.789642\pi\)
−0.789465 + 0.613796i \(0.789642\pi\)
\(84\) 1.30789e15 0.0575701
\(85\) 3.05154e15 0.121467
\(86\) −1.98855e16 −0.716638
\(87\) 1.07083e15 0.0349790
\(88\) 7.15343e15 0.212037
\(89\) −4.91787e16 −1.32423 −0.662113 − 0.749404i \(-0.730340\pi\)
−0.662113 + 0.749404i \(0.730340\pi\)
\(90\) 5.82605e15 0.142664
\(91\) 5.89681e16 1.31451
\(92\) −1.84316e15 −0.0374427
\(93\) −3.53484e16 −0.655036
\(94\) 4.93784e16 0.835507
\(95\) −2.80531e16 −0.433841
\(96\) 6.84677e15 0.0968680
\(97\) 1.26124e17 1.63394 0.816972 − 0.576678i \(-0.195651\pi\)
0.816972 + 0.576678i \(0.195651\pi\)
\(98\) 3.26543e16 0.387720
\(99\) −6.25450e15 −0.0681230
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 15.18.a.a.1.2 ✓ 2
3.2 odd 2 45.18.a.b.1.1 2
5.2 odd 4 75.18.b.b.49.3 4
5.3 odd 4 75.18.b.b.49.2 4
5.4 even 2 75.18.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.a.1.2 ✓ 2 1.1 even 1 trivial
45.18.a.b.1.1 2 3.2 odd 2
75.18.a.c.1.1 2 5.4 even 2
75.18.b.b.49.2 4 5.3 odd 4
75.18.b.b.49.3 4 5.2 odd 4