Properties

Label 300.8.a.f.1.1
Level $300$
Weight $8$
Character 300.1
Self dual yes
Analytic conductor $93.716$
Analytic rank $1$
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: \(1\)
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 \(-0.369774\pi\)
0.397799 + 0.917472i \(0.369774\pi\)
\(8\) 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 \(-0.561255\pi\)
−0.191254 + 0.981541i \(0.561255\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 20582.0 1.01605 0.508026 − 0.861341i \(-0.330375\pi\)
0.508026 + 0.861341i \(0.330375\pi\)
\(18\) 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 \(-0.693495\pi\)
−0.571131 + 0.820859i \(0.693495\pi\)
\(24\) 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 \(-0.315268\pi\)
0.548320 + 0.836269i \(0.315268\pi\)
\(38\) 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 \(-0.639623\pi\)
−0.424707 + 0.905331i \(0.639623\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −170432. −0.239447 −0.119723 − 0.992807i \(-0.538201\pi\)
−0.119723 + 0.992807i \(0.538201\pi\)
\(48\) 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 \(-0.306255\pi\)
0.571775 + 0.820411i \(0.306255\pi\)
\(54\) 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 \(-0.774727\pi\)
−0.759849 + 0.650099i \(0.774727\pi\)
\(68\) 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 \(-0.617815\pi\)
−0.361734 + 0.932281i \(0.617815\pi\)
\(74\) 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 \(-0.330135\pi\)
0.508677 + 0.860957i \(0.330135\pi\)
\(84\) 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 \(-0.311047\pi\)
0.559360 + 0.828925i \(0.311047\pi\)
\(98\) 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.f.1.1 1
5.2 odd 4 60.8.d.a.49.1 ✓ 2
5.3 odd 4 60.8.d.a.49.2 yes 2
5.4 even 2 300.8.a.b.1.1 1
15.2 even 4 180.8.d.a.109.1 2
15.8 even 4 180.8.d.a.109.2 2
20.3 even 4 240.8.f.b.49.1 2
20.7 even 4 240.8.f.b.49.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.8.d.a.49.1 ✓ 2 5.2 odd 4
60.8.d.a.49.2 yes 2 5.3 odd 4
180.8.d.a.109.1 2 15.2 even 4
180.8.d.a.109.2 2 15.8 even 4
240.8.f.b.49.1 2 20.3 even 4
240.8.f.b.49.2 2 20.7 even 4
300.8.a.b.1.1 1 5.4 even 2
300.8.a.f.1.1 1 1.1 even 1 trivial