Properties

Label 300.8.a.b.1.1
Level $300$
Weight $8$
Character 300.1
Self dual yes
Analytic conductor $93.716$
Analytic rank $0$
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] = [300,8,Mod(1,300)] 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("300.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 300.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-27,0,0,0,-722,0,729,0,-3994] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(93.7155076452\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 300.1

$q$-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)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} -722.000 q^{7} +729.000 q^{9} -3994.00 q^{11} +3030.00 q^{13} -20582.0 q^{17} -25320.0 q^{19} +19494.0 q^{21} +66652.0 q^{23} -19683.0 q^{27} -152664. q^{29} -123776. q^{31} +107838. q^{33} -337886. q^{37} -81810.0 q^{39} +396530. q^{41} +442852. q^{43} +170432. q^{47} -302259. q^{49} +555714. q^{51} -1.23943e6 q^{53} +683640. q^{57} -302354. q^{59} -2.83020e6 q^{61} -526338. q^{63} +3.74127e6 q^{67} -1.79960e6 q^{69} -1.00758e6 q^{71} +2.40464e6 q^{73} +2.88367e6 q^{77} +7.51783e6 q^{79} +531441. q^{81} -5.29963e6 q^{83} +4.12193e6 q^{87} -7.65025e6 q^{89} -2.18766e6 q^{91} +3.34195e6 q^{93} -1.00559e7 q^{97} -2.91163e6 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −27.0000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −722.000 −0.795599 −0.397799 − 0.917472i \(-0.630226\pi\)
−0.397799 + 0.917472i \(0.630226\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −3994.00 −0.904761 −0.452380 − 0.891825i \(-0.649425\pi\)
−0.452380 + 0.891825i \(0.649425\pi\)
\(12\) 0 0
\(13\) 3030.00 0.382508 0.191254 − 0.981541i \(-0.438745\pi\)
0.191254 + 0.981541i \(0.438745\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −20582.0 −1.01605 −0.508026 − 0.861341i \(-0.669625\pi\)
−0.508026 + 0.861341i \(0.669625\pi\)
\(18\) 0 0
\(19\) −25320.0 −0.846888 −0.423444 − 0.905922i \(-0.639179\pi\)
−0.423444 + 0.905922i \(0.639179\pi\)
\(20\) 0 0
\(21\) 19494.0 0.459339
\(22\) 0 0
\(23\) 66652.0 1.14226 0.571131 − 0.820859i \(-0.306505\pi\)
0.571131 + 0.820859i \(0.306505\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) −152664. −1.16237 −0.581184 − 0.813772i \(-0.697410\pi\)
−0.581184 + 0.813772i \(0.697410\pi\)
\(30\) 0 0
\(31\) −123776. −0.746226 −0.373113 − 0.927786i \(-0.621710\pi\)
−0.373113 + 0.927786i \(0.621710\pi\)
\(32\) 0 0
\(33\) 107838. 0.522364
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −337886. −1.09664 −0.548320 − 0.836269i \(-0.684732\pi\)
−0.548320 + 0.836269i \(0.684732\pi\)
\(38\) 0 0
\(39\) −81810.0 −0.220841
\(40\) 0 0
\(41\) 396530. 0.898530 0.449265 − 0.893399i \(-0.351686\pi\)
0.449265 + 0.893399i \(0.351686\pi\)
\(42\) 0 0
\(43\) 442852. 0.849413 0.424707 − 0.905331i \(-0.360377\pi\)
0.424707 + 0.905331i \(0.360377\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 170432. 0.239447 0.119723 − 0.992807i \(-0.461799\pi\)
0.119723 + 0.992807i \(0.461799\pi\)
\(48\) 0 0
\(49\) −302259. −0.367023
\(50\) 0 0
\(51\) 555714. 0.586618
\(52\) 0 0
\(53\) −1.23943e6 −1.14355 −0.571775 − 0.820411i \(-0.693745\pi\)
−0.571775 + 0.820411i \(0.693745\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 683640. 0.488951
\(58\) 0 0
\(59\) −302354. −0.191661 −0.0958305 − 0.995398i \(-0.530551\pi\)
−0.0958305 + 0.995398i \(0.530551\pi\)
\(60\) 0 0
\(61\) −2.83020e6 −1.59648 −0.798238 − 0.602342i \(-0.794234\pi\)
−0.798238 + 0.602342i \(0.794234\pi\)
\(62\) 0 0
\(63\) −526338. −0.265200
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.74127e6 1.51970 0.759849 − 0.650099i \(-0.225273\pi\)
0.759849 + 0.650099i \(0.225273\pi\)
\(68\) 0 0
\(69\) −1.79960e6 −0.659485
\(70\) 0 0
\(71\) −1.00758e6 −0.334099 −0.167050 − 0.985949i \(-0.553424\pi\)
−0.167050 + 0.985949i \(0.553424\pi\)
\(72\) 0 0
\(73\) 2.40464e6 0.723468 0.361734 − 0.932281i \(-0.382185\pi\)
0.361734 + 0.932281i \(0.382185\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.88367e6 0.719826
\(78\) 0 0
\(79\) 7.51783e6 1.71553 0.857764 − 0.514044i \(-0.171853\pi\)
0.857764 + 0.514044i \(0.171853\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) −5.29963e6 −1.01735 −0.508677 − 0.860957i \(-0.669865\pi\)
−0.508677 + 0.860957i \(0.669865\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.12193e6 0.671093
\(88\) 0 0
\(89\) −7.65025e6 −1.15030 −0.575149 − 0.818048i \(-0.695056\pi\)
−0.575149 + 0.818048i \(0.695056\pi\)
\(90\) 0 0
\(91\) −2.18766e6 −0.304323
\(92\) 0 0
\(93\) 3.34195e6 0.430834
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.00559e7 −1.11872 −0.559360 − 0.828925i \(-0.688953\pi\)
−0.559360 + 0.828925i \(0.688953\pi\)
\(98\) 0 0
\(99\) −2.91163e6 −0.301587
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 300.8.a.b.1.1 1
5.2 odd 4 60.8.d.a.49.2 yes 2
5.3 odd 4 60.8.d.a.49.1 ✓ 2
5.4 even 2 300.8.a.f.1.1 1
15.2 even 4 180.8.d.a.109.2 2
15.8 even 4 180.8.d.a.109.1 2
20.3 even 4 240.8.f.b.49.2 2
20.7 even 4 240.8.f.b.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.8.d.a.49.1 ✓ 2 5.3 odd 4
60.8.d.a.49.2 yes 2 5.2 odd 4
180.8.d.a.109.1 2 15.8 even 4
180.8.d.a.109.2 2 15.2 even 4
240.8.f.b.49.1 2 20.7 even 4
240.8.f.b.49.2 2 20.3 even 4
300.8.a.b.1.1 1 1.1 even 1 trivial
300.8.a.f.1.1 1 5.4 even 2