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Properties

Label 45.18.a.d.1.3
Level $45$
Weight $18$
Character 45.1
Self dual yes
Analytic conductor $82.450$
Analytic rank $1$
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] = [45,18,Mod(1,45)] 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("45.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 45.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: \(82.4499393051\)
Analytic rank: \(1\)
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: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(416.408\) of defining polynomial
Character \(\chi\) \(=\) 45.1

$q$-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)
 
\(f(q)\) \(=\) \(q+500.408 q^{2} +119336. q^{4} +390625. q^{5} +8.90512e6 q^{7} -5.87255e6 q^{8} +1.95472e8 q^{10} -4.48684e8 q^{11} -3.89738e9 q^{13} +4.45620e9 q^{14} -1.85803e10 q^{16} +5.96526e8 q^{17} -4.36939e10 q^{19} +4.66158e10 q^{20} -2.24525e11 q^{22} -7.23343e10 q^{23} +1.52588e11 q^{25} -1.95028e12 q^{26} +1.06271e12 q^{28} -1.82054e12 q^{29} +5.27128e12 q^{31} -8.52803e12 q^{32} +2.98507e11 q^{34} +3.47856e12 q^{35} -1.62659e13 q^{37} -2.18648e13 q^{38} -2.29397e12 q^{40} -9.78891e12 q^{41} +1.46155e14 q^{43} -5.35444e13 q^{44} -3.61967e13 q^{46} -2.43226e14 q^{47} -1.53329e14 q^{49} +7.63562e13 q^{50} -4.65100e14 q^{52} -6.84721e14 q^{53} -1.75267e14 q^{55} -5.22958e13 q^{56} -9.11015e14 q^{58} -9.83635e14 q^{59} -1.84021e15 q^{61} +2.63779e15 q^{62} -1.83214e15 q^{64} -1.52241e15 q^{65} -2.46536e15 q^{67} +7.11874e13 q^{68} +1.74070e15 q^{70} -9.43154e14 q^{71} +1.15522e16 q^{73} -8.13959e15 q^{74} -5.21428e15 q^{76} -3.99559e15 q^{77} +1.58855e16 q^{79} -7.25795e15 q^{80} -4.89845e15 q^{82} -1.62194e16 q^{83} +2.33018e14 q^{85} +7.31369e16 q^{86} +2.63492e15 q^{88} +4.03683e16 q^{89} -3.47067e16 q^{91} -8.63212e15 q^{92} -1.21712e17 q^{94} -1.70679e16 q^{95} -7.85724e16 q^{97} -7.67273e16 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 253 q^{2} - 7087 q^{4} + 1171875 q^{5} - 4332484 q^{7} + 16188513 q^{8} + 98828125 q^{10} - 943563680 q^{11} + 4257013150 q^{13} - 1847483988 q^{14} - 21943871359 q^{16} - 3063169514 q^{17} - 78122492996 q^{19}+ \cdots + 896231490630245 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 500.408 1.38220 0.691098 − 0.722761i \(-0.257127\pi\)
0.691098 + 0.722761i \(0.257127\pi\)
\(3\) 0 0
\(4\) 119336. 0.910465
\(5\) 390625. 0.447214
\(6\) 0 0
\(7\) 8.90512e6 0.583857 0.291929 − 0.956440i \(-0.405703\pi\)
0.291929 + 0.956440i \(0.405703\pi\)
\(8\) −5.87255e6 −0.123755
\(9\) 0 0
\(10\) 1.95472e8 0.618137
\(11\) −4.48684e8 −0.631107 −0.315554 − 0.948908i \(-0.602190\pi\)
−0.315554 + 0.948908i \(0.602190\pi\)
\(12\) 0 0
\(13\) −3.89738e9 −1.32512 −0.662559 − 0.749010i \(-0.730530\pi\)
−0.662559 + 0.749010i \(0.730530\pi\)
\(14\) 4.45620e9 0.807005
\(15\) 0 0
\(16\) −1.85803e10 −1.08152
\(17\) 5.96526e8 0.0207402 0.0103701 − 0.999946i \(-0.496699\pi\)
0.0103701 + 0.999946i \(0.496699\pi\)
\(18\) 0 0
\(19\) −4.36939e10 −0.590222 −0.295111 − 0.955463i \(-0.595357\pi\)
−0.295111 + 0.955463i \(0.595357\pi\)
\(20\) 4.66158e10 0.407172
\(21\) 0 0
\(22\) −2.24525e11 −0.872314
\(23\) −7.23343e10 −0.192601 −0.0963003 − 0.995352i \(-0.530701\pi\)
−0.0963003 + 0.995352i \(0.530701\pi\)
\(24\) 0 0
\(25\) 1.52588e11 0.200000
\(26\) −1.95028e12 −1.83157
\(27\) 0 0
\(28\) 1.06271e12 0.531582
\(29\) −1.82054e12 −0.675800 −0.337900 − 0.941182i \(-0.609716\pi\)
−0.337900 + 0.941182i \(0.609716\pi\)
\(30\) 0 0
\(31\) 5.27128e12 1.11005 0.555024 − 0.831835i \(-0.312709\pi\)
0.555024 + 0.831835i \(0.312709\pi\)
\(32\) −8.52803e12 −1.37112
\(33\) 0 0
\(34\) 2.98507e11 0.0286671
\(35\) 3.47856e12 0.261109
\(36\) 0 0
\(37\) −1.62659e13 −0.761313 −0.380656 − 0.924717i \(-0.624302\pi\)
−0.380656 + 0.924717i \(0.624302\pi\)
\(38\) −2.18648e13 −0.815803
\(39\) 0 0
\(40\) −2.29397e12 −0.0553448
\(41\) −9.78891e12 −0.191457 −0.0957285 − 0.995407i \(-0.530518\pi\)
−0.0957285 + 0.995407i \(0.530518\pi\)
\(42\) 0 0
\(43\) 1.46155e14 1.90691 0.953455 − 0.301534i \(-0.0974987\pi\)
0.953455 + 0.301534i \(0.0974987\pi\)
\(44\) −5.35444e13 −0.574601
\(45\) 0 0
\(46\) −3.61967e13 −0.266212
\(47\) −2.43226e14 −1.48997 −0.744987 − 0.667079i \(-0.767544\pi\)
−0.744987 + 0.667079i \(0.767544\pi\)
\(48\) 0 0
\(49\) −1.53329e14 −0.659111
\(50\) 7.63562e13 0.276439
\(51\) 0 0
\(52\) −4.65100e14 −1.20647
\(53\) −6.84721e14 −1.51067 −0.755334 − 0.655340i \(-0.772525\pi\)
−0.755334 + 0.655340i \(0.772525\pi\)
\(54\) 0 0
\(55\) −1.75267e14 −0.282240
\(56\) −5.22958e13 −0.0722551
\(57\) 0 0
\(58\) −9.11015e14 −0.934087
\(59\) −9.83635e14 −0.872151 −0.436076 − 0.899910i \(-0.643632\pi\)
−0.436076 + 0.899910i \(0.643632\pi\)
\(60\) 0 0
\(61\) −1.84021e15 −1.22903 −0.614516 − 0.788904i \(-0.710649\pi\)
−0.614516 + 0.788904i \(0.710649\pi\)
\(62\) 2.63779e15 1.53430
\(63\) 0 0
\(64\) −1.83214e15 −0.813631
\(65\) −1.52241e15 −0.592611
\(66\) 0 0
\(67\) −2.46536e15 −0.741729 −0.370864 − 0.928687i \(-0.620938\pi\)
−0.370864 + 0.928687i \(0.620938\pi\)
\(68\) 7.11874e13 0.0188833
\(69\) 0 0
\(70\) 1.74070e15 0.360904
\(71\) −9.43154e14 −0.173335 −0.0866676 − 0.996237i \(-0.527622\pi\)
−0.0866676 + 0.996237i \(0.527622\pi\)
\(72\) 0 0
\(73\) 1.15522e16 1.67657 0.838286 − 0.545231i \(-0.183558\pi\)
0.838286 + 0.545231i \(0.183558\pi\)
\(74\) −8.13959e15 −1.05228
\(75\) 0 0
\(76\) −5.21428e15 −0.537377
\(77\) −3.99559e15 −0.368477
\(78\) 0 0
\(79\) 1.58855e16 1.17807 0.589035 − 0.808108i \(-0.299508\pi\)
0.589035 + 0.808108i \(0.299508\pi\)
\(80\) −7.25795e15 −0.483670
\(81\) 0 0
\(82\) −4.89845e15 −0.264631
\(83\) −1.62194e16 −0.790442 −0.395221 − 0.918586i \(-0.629332\pi\)
−0.395221 + 0.918586i \(0.629332\pi\)
\(84\) 0 0
\(85\) 2.33018e14 0.00927532
\(86\) 7.31369e16 2.63572
\(87\) 0 0
\(88\) 2.63492e15 0.0781026
\(89\) 4.03683e16 1.08699 0.543495 − 0.839413i \(-0.317101\pi\)
0.543495 + 0.839413i \(0.317101\pi\)
\(90\) 0 0
\(91\) −3.47067e16 −0.773679
\(92\) −8.63212e15 −0.175356
\(93\) 0 0
\(94\) −1.21712e17 −2.05944
\(95\) −1.70679e16 −0.263955
\(96\) 0 0
\(97\) −7.85724e16 −1.01791 −0.508956 − 0.860792i \(-0.669969\pi\)
−0.508956 + 0.860792i \(0.669969\pi\)
\(98\) −7.67273e16 −0.911020
\(99\) 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 45.18.a.d.1.3 3
3.2 odd 2 15.18.a.c.1.1 ✓ 3
15.2 even 4 75.18.b.e.49.1 6
15.8 even 4 75.18.b.e.49.6 6
15.14 odd 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 3.2 odd 2
45.18.a.d.1.3 3 1.1 even 1 trivial
75.18.a.d.1.3 3 15.14 odd 2
75.18.b.e.49.1 6 15.2 even 4
75.18.b.e.49.6 6 15.8 even 4