Properties

Label 300.8.a.h.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 = [1,0,27,0,0,0,832,0,729,0,3156] 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)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} +832.000 q^{7} +729.000 q^{9} +3156.00 q^{11} +7690.00 q^{13} -258.000 q^{17} +45740.0 q^{19} +22464.0 q^{21} -104832. q^{23} +19683.0 q^{27} +38646.0 q^{29} +192224. q^{31} +85212.0 q^{33} -403454. q^{37} +207630. q^{39} +86010.0 q^{41} +127348. q^{43} -601272. q^{47} -131319. q^{49} -6966.00 q^{51} +1.62823e6 q^{53} +1.23498e6 q^{57} +198996. q^{59} +1.20978e6 q^{61} +606528. q^{63} +699388. q^{67} -2.83046e6 q^{69} -4.93932e6 q^{71} +1.27533e6 q^{73} +2.62579e6 q^{77} +6.55971e6 q^{79} +531441. q^{81} +3.10835e6 q^{83} +1.04344e6 q^{87} +5.54241e6 q^{89} +6.39808e6 q^{91} +5.19005e6 q^{93} -4.51335e6 q^{97} +2.30072e6 q^{99} +O(q^{100})\)

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
\(7\) 832.000 0.916812 0.458406 − 0.888743i \(-0.348421\pi\)
0.458406 + 0.888743i \(0.348421\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 3156.00 0.714928 0.357464 − 0.933927i \(-0.383641\pi\)
0.357464 + 0.933927i \(0.383641\pi\)
\(12\) 0 0
\(13\) 7690.00 0.970788 0.485394 − 0.874295i \(-0.338676\pi\)
0.485394 + 0.874295i \(0.338676\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −258.000 −0.0127365 −0.00636823 − 0.999980i \(-0.502027\pi\)
−0.00636823 + 0.999980i \(0.502027\pi\)
\(18\) 0 0
\(19\) 45740.0 1.52988 0.764942 − 0.644099i \(-0.222768\pi\)
0.764942 + 0.644099i \(0.222768\pi\)
\(20\) 0 0
\(21\) 22464.0 0.529322
\(22\) 0 0
\(23\) −104832. −1.79658 −0.898290 − 0.439404i \(-0.855190\pi\)
−0.898290 + 0.439404i \(0.855190\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) 38646.0 0.294247 0.147123 − 0.989118i \(-0.452999\pi\)
0.147123 + 0.989118i \(0.452999\pi\)
\(30\) 0 0
\(31\) 192224. 1.15889 0.579444 − 0.815012i \(-0.303270\pi\)
0.579444 + 0.815012i \(0.303270\pi\)
\(32\) 0 0
\(33\) 85212.0 0.412764
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −403454. −1.30945 −0.654724 − 0.755868i \(-0.727215\pi\)
−0.654724 + 0.755868i \(0.727215\pi\)
\(38\) 0 0
\(39\) 207630. 0.560485
\(40\) 0 0
\(41\) 86010.0 0.194897 0.0974486 − 0.995241i \(-0.468932\pi\)
0.0974486 + 0.995241i \(0.468932\pi\)
\(42\) 0 0
\(43\) 127348. 0.244260 0.122130 − 0.992514i \(-0.461028\pi\)
0.122130 + 0.992514i \(0.461028\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −601272. −0.844751 −0.422375 − 0.906421i \(-0.638804\pi\)
−0.422375 + 0.906421i \(0.638804\pi\)
\(48\) 0 0
\(49\) −131319. −0.159456
\(50\) 0 0
\(51\) −6966.00 −0.00735339
\(52\) 0 0
\(53\) 1.62823e6 1.50227 0.751137 − 0.660146i \(-0.229506\pi\)
0.751137 + 0.660146i \(0.229506\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.23498e6 0.883279
\(58\) 0 0
\(59\) 198996. 0.126143 0.0630714 − 0.998009i \(-0.479910\pi\)
0.0630714 + 0.998009i \(0.479910\pi\)
\(60\) 0 0
\(61\) 1.20978e6 0.682421 0.341211 − 0.939987i \(-0.389163\pi\)
0.341211 + 0.939987i \(0.389163\pi\)
\(62\) 0 0
\(63\) 606528. 0.305604
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 699388. 0.284090 0.142045 − 0.989860i \(-0.454632\pi\)
0.142045 + 0.989860i \(0.454632\pi\)
\(68\) 0 0
\(69\) −2.83046e6 −1.03726
\(70\) 0 0
\(71\) −4.93932e6 −1.63781 −0.818904 − 0.573931i \(-0.805418\pi\)
−0.818904 + 0.573931i \(0.805418\pi\)
\(72\) 0 0
\(73\) 1.27533e6 0.383702 0.191851 − 0.981424i \(-0.438551\pi\)
0.191851 + 0.981424i \(0.438551\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.62579e6 0.655455
\(78\) 0 0
\(79\) 6.55971e6 1.49689 0.748445 − 0.663197i \(-0.230801\pi\)
0.748445 + 0.663197i \(0.230801\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 3.10835e6 0.596700 0.298350 − 0.954456i \(-0.403564\pi\)
0.298350 + 0.954456i \(0.403564\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.04344e6 0.169883
\(88\) 0 0
\(89\) 5.54241e6 0.833362 0.416681 − 0.909053i \(-0.363193\pi\)
0.416681 + 0.909053i \(0.363193\pi\)
\(90\) 0 0
\(91\) 6.39808e6 0.890030
\(92\) 0 0
\(93\) 5.19005e6 0.669085
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.51335e6 −0.502108 −0.251054 − 0.967973i \(-0.580777\pi\)
−0.251054 + 0.967973i \(0.580777\pi\)
\(98\) 0 0
\(99\) 2.30072e6 0.238309
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.h.1.1 1
5.2 odd 4 300.8.d.e.49.1 2
5.3 odd 4 300.8.d.e.49.2 2
5.4 even 2 60.8.a.b.1.1 ✓ 1
15.14 odd 2 180.8.a.b.1.1 1
20.19 odd 2 240.8.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.8.a.b.1.1 ✓ 1 5.4 even 2
180.8.a.b.1.1 1 15.14 odd 2
240.8.a.n.1.1 1 20.19 odd 2
300.8.a.h.1.1 1 1.1 even 1 trivial
300.8.d.e.49.1 2 5.2 odd 4
300.8.d.e.49.2 2 5.3 odd 4