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Properties

Label 15.18.a.c.1.1
Level $15$
Weight $18$
Character 15.1
Self dual yes
Analytic conductor $27.483$
Analytic rank $0$
Dimension $3$
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 = [3,-253] 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: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 182396x + 3921120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(416.408\) 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)
 
\(f(q)\) \(=\) \(q-500.408 q^{2} +6561.00 q^{3} +119336. q^{4} -390625. q^{5} -3.28318e6 q^{6} +8.90512e6 q^{7} +5.87255e6 q^{8} +4.30467e7 q^{9} +1.95472e8 q^{10} +4.48684e8 q^{11} +7.82967e8 q^{12} -3.89738e9 q^{13} -4.45620e9 q^{14} -2.56289e9 q^{15} -1.85803e10 q^{16} -5.96526e8 q^{17} -2.15409e10 q^{18} -4.36939e10 q^{19} -4.66158e10 q^{20} +5.84265e10 q^{21} -2.24525e11 q^{22} +7.23343e10 q^{23} +3.85298e10 q^{24} +1.52588e11 q^{25} +1.95028e12 q^{26} +2.82430e11 q^{27} +1.06271e12 q^{28} +1.82054e12 q^{29} +1.28249e12 q^{30} +5.27128e12 q^{31} +8.52803e12 q^{32} +2.94382e12 q^{33} +2.98507e11 q^{34} -3.47856e12 q^{35} +5.13704e12 q^{36} -1.62659e13 q^{37} +2.18648e13 q^{38} -2.55707e13 q^{39} -2.29397e12 q^{40} +9.78891e12 q^{41} -2.92371e13 q^{42} +1.46155e14 q^{43} +5.35444e13 q^{44} -1.68151e13 q^{45} -3.61967e13 q^{46} +2.43226e14 q^{47} -1.21906e14 q^{48} -1.53329e14 q^{49} -7.63562e13 q^{50} -3.91381e12 q^{51} -4.65100e14 q^{52} +6.84721e14 q^{53} -1.41330e14 q^{54} -1.75267e14 q^{55} +5.22958e13 q^{56} -2.86676e14 q^{57} -9.11015e14 q^{58} +9.83635e14 q^{59} -3.05846e14 q^{60} -1.84021e15 q^{61} -2.63779e15 q^{62} +3.83336e14 q^{63} -1.83214e15 q^{64} +1.52241e15 q^{65} -1.47311e15 q^{66} -2.46536e15 q^{67} -7.11874e13 q^{68} +4.74585e14 q^{69} +1.74070e15 q^{70} +9.43154e14 q^{71} +2.52794e14 q^{72} +1.15522e16 q^{73} +8.13959e15 q^{74} +1.00113e15 q^{75} -5.21428e15 q^{76} +3.99559e15 q^{77} +1.27958e16 q^{78} +1.58855e16 q^{79} +7.25795e15 q^{80} +1.85302e15 q^{81} -4.89845e15 q^{82} +1.62194e16 q^{83} +6.97241e15 q^{84} +2.33018e14 q^{85} -7.31369e16 q^{86} +1.19446e16 q^{87} +2.63492e15 q^{88} -4.03683e16 q^{89} +8.41443e15 q^{90} -3.47067e16 q^{91} +8.63212e15 q^{92} +3.45849e16 q^{93} -1.21712e17 q^{94} +1.70679e16 q^{95} +5.59524e16 q^{96} -7.85724e16 q^{97} +7.67273e16 q^{98} +1.93144e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 253 q^{2} + 19683 q^{3} - 7087 q^{4} - 1171875 q^{5} - 1659933 q^{6} - 4332484 q^{7} - 16188513 q^{8} + 129140163 q^{9} + 98828125 q^{10} + 943563680 q^{11} - 46497807 q^{12} + 4257013150 q^{13} + 1847483988 q^{14}+ \cdots + 40\!\cdots\!80 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −500.408 −1.38220 −0.691098 − 0.722761i \(-0.742873\pi\)
−0.691098 + 0.722761i \(0.742873\pi\)
\(3\) 6561.00 0.577350
\(4\) 119336. 0.910465
\(5\) −390625. −0.447214
\(6\) −3.28318e6 −0.798011
\(7\) 8.90512e6 0.583857 0.291929 − 0.956440i \(-0.405703\pi\)
0.291929 + 0.956440i \(0.405703\pi\)
\(8\) 5.87255e6 0.123755
\(9\) 4.30467e7 0.333333
\(10\) 1.95472e8 0.618137
\(11\) 4.48684e8 0.631107 0.315554 − 0.948908i \(-0.397810\pi\)
0.315554 + 0.948908i \(0.397810\pi\)
\(12\) 7.82967e8 0.525657
\(13\) −3.89738e9 −1.32512 −0.662559 − 0.749010i \(-0.730530\pi\)
−0.662559 + 0.749010i \(0.730530\pi\)
\(14\) −4.45620e9 −0.807005
\(15\) −2.56289e9 −0.258199
\(16\) −1.85803e10 −1.08152
\(17\) −5.96526e8 −0.0207402 −0.0103701 − 0.999946i \(-0.503301\pi\)
−0.0103701 + 0.999946i \(0.503301\pi\)
\(18\) −2.15409e10 −0.460732
\(19\) −4.36939e10 −0.590222 −0.295111 − 0.955463i \(-0.595357\pi\)
−0.295111 + 0.955463i \(0.595357\pi\)
\(20\) −4.66158e10 −0.407172
\(21\) 5.84265e10 0.337090
\(22\) −2.24525e11 −0.872314
\(23\) 7.23343e10 0.192601 0.0963003 − 0.995352i \(-0.469299\pi\)
0.0963003 + 0.995352i \(0.469299\pi\)
\(24\) 3.85298e10 0.0714499
\(25\) 1.52588e11 0.200000
\(26\) 1.95028e12 1.83157
\(27\) 2.82430e11 0.192450
\(28\) 1.06271e12 0.531582
\(29\) 1.82054e12 0.675800 0.337900 − 0.941182i \(-0.390284\pi\)
0.337900 + 0.941182i \(0.390284\pi\)
\(30\) 1.28249e12 0.356881
\(31\) 5.27128e12 1.11005 0.555024 − 0.831835i \(-0.312709\pi\)
0.555024 + 0.831835i \(0.312709\pi\)
\(32\) 8.52803e12 1.37112
\(33\) 2.94382e12 0.364370
\(34\) 2.98507e11 0.0286671
\(35\) −3.47856e12 −0.261109
\(36\) 5.13704e12 0.303488
\(37\) −1.62659e13 −0.761313 −0.380656 − 0.924717i \(-0.624302\pi\)
−0.380656 + 0.924717i \(0.624302\pi\)
\(38\) 2.18648e13 0.815803
\(39\) −2.55707e13 −0.765057
\(40\) −2.29397e12 −0.0553448
\(41\) 9.78891e12 0.191457 0.0957285 − 0.995407i \(-0.469482\pi\)
0.0957285 + 0.995407i \(0.469482\pi\)
\(42\) −2.92371e13 −0.465924
\(43\) 1.46155e14 1.90691 0.953455 − 0.301534i \(-0.0974987\pi\)
0.953455 + 0.301534i \(0.0974987\pi\)
\(44\) 5.35444e13 0.574601
\(45\) −1.68151e13 −0.149071
\(46\) −3.61967e13 −0.266212
\(47\) 2.43226e14 1.48997 0.744987 − 0.667079i \(-0.232456\pi\)
0.744987 + 0.667079i \(0.232456\pi\)
\(48\) −1.21906e14 −0.624415
\(49\) −1.53329e14 −0.659111
\(50\) −7.63562e13 −0.276439
\(51\) −3.91381e12 −0.0119744
\(52\) −4.65100e14 −1.20647
\(53\) 6.84721e14 1.51067 0.755334 − 0.655340i \(-0.227475\pi\)
0.755334 + 0.655340i \(0.227475\pi\)
\(54\) −1.41330e14 −0.266004
\(55\) −1.75267e14 −0.282240
\(56\) 5.22958e13 0.0722551
\(57\) −2.86676e14 −0.340765
\(58\) −9.11015e14 −0.934087
\(59\) 9.83635e14 0.872151 0.436076 − 0.899910i \(-0.356368\pi\)
0.436076 + 0.899910i \(0.356368\pi\)
\(60\) −3.05846e14 −0.235081
\(61\) −1.84021e15 −1.22903 −0.614516 − 0.788904i \(-0.710649\pi\)
−0.614516 + 0.788904i \(0.710649\pi\)
\(62\) −2.63779e15 −1.53430
\(63\) 3.83336e14 0.194619
\(64\) −1.83214e15 −0.813631
\(65\) 1.52241e15 0.592611
\(66\) −1.47311e15 −0.503631
\(67\) −2.46536e15 −0.741729 −0.370864 − 0.928687i \(-0.620938\pi\)
−0.370864 + 0.928687i \(0.620938\pi\)
\(68\) −7.11874e13 −0.0188833
\(69\) 4.74585e14 0.111198
\(70\) 1.74070e15 0.360904
\(71\) 9.43154e14 0.173335 0.0866676 − 0.996237i \(-0.472378\pi\)
0.0866676 + 0.996237i \(0.472378\pi\)
\(72\) 2.52794e14 0.0412516
\(73\) 1.15522e16 1.67657 0.838286 − 0.545231i \(-0.183558\pi\)
0.838286 + 0.545231i \(0.183558\pi\)
\(74\) 8.13959e15 1.05228
\(75\) 1.00113e15 0.115470
\(76\) −5.21428e15 −0.537377
\(77\) 3.99559e15 0.368477
\(78\) 1.27958e16 1.05746
\(79\) 1.58855e16 1.17807 0.589035 − 0.808108i \(-0.299508\pi\)
0.589035 + 0.808108i \(0.299508\pi\)
\(80\) 7.25795e15 0.483670
\(81\) 1.85302e15 0.111111
\(82\) −4.89845e15 −0.264631
\(83\) 1.62194e16 0.790442 0.395221 − 0.918586i \(-0.370668\pi\)
0.395221 + 0.918586i \(0.370668\pi\)
\(84\) 6.97241e15 0.306909
\(85\) 2.33018e14 0.00927532
\(86\) −7.31369e16 −2.63572
\(87\) 1.19446e16 0.390173
\(88\) 2.63492e15 0.0781026
\(89\) −4.03683e16 −1.08699 −0.543495 − 0.839413i \(-0.682899\pi\)
−0.543495 + 0.839413i \(0.682899\pi\)
\(90\) 8.41443e15 0.206046
\(91\) −3.47067e16 −0.773679
\(92\) 8.63212e15 0.175356
\(93\) 3.45849e16 0.640886
\(94\) −1.21712e17 −2.05944
\(95\) 1.70679e16 0.263955
\(96\) 5.59524e16 0.791614
\(97\) −7.85724e16 −1.01791 −0.508956 − 0.860792i \(-0.669969\pi\)
−0.508956 + 0.860792i \(0.669969\pi\)
\(98\) 7.67273e16 0.911020
\(99\) 1.93144e16 0.210369
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.c.1.1 ✓ 3
3.2 odd 2 45.18.a.d.1.3 3
5.2 odd 4 75.18.b.e.49.1 6
5.3 odd 4 75.18.b.e.49.6 6
5.4 even 2 75.18.a.d.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.c.1.1 ✓ 3 1.1 even 1 trivial
45.18.a.d.1.3 3 3.2 odd 2
75.18.a.d.1.3 3 5.4 even 2
75.18.b.e.49.1 6 5.2 odd 4
75.18.b.e.49.6 6 5.3 odd 4