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 = [3,0,81,0,0,0,351,0,2187,0,3138] 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: \(3\)
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: \( x^{3} - x^{2} - 572x - 3990 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-18.3302\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} -1002.81 q^{7} +729.000 q^{9} +6311.33 q^{11} -11909.0 q^{13} +288.453 q^{17} +28007.7 q^{19} -27075.9 q^{21} -28288.8 q^{23} +19683.0 q^{27} +97664.9 q^{29} -290434. q^{31} +170406. q^{33} +24738.0 q^{37} -321543. q^{39} +688629. q^{41} -396038. q^{43} +920149. q^{47} +182089. q^{49} +7788.24 q^{51} +1.64832e6 q^{53} +756208. q^{57} +2.34227e6 q^{59} -128675. q^{61} -731050. q^{63} +2.69575e6 q^{67} -763797. q^{69} -2.06763e6 q^{71} -2.11158e6 q^{73} -6.32907e6 q^{77} -694941. q^{79} +531441. q^{81} -4.03690e6 q^{83} +2.63695e6 q^{87} +3.43798e6 q^{89} +1.19425e7 q^{91} -7.84172e6 q^{93} +7.90999e6 q^{97} +4.60096e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 81 q^{3} + 351 q^{7} + 2187 q^{9} + 3138 q^{11} + 3585 q^{13} + 5226 q^{17} + 21195 q^{19} + 9477 q^{21} + 40434 q^{23} + 59049 q^{27} + 26598 q^{29} - 143463 q^{31} + 84726 q^{33} + 536658 q^{37}+ \cdots + 2287602 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 27.0000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1002.81 −1.10504 −0.552518 − 0.833501i \(-0.686333\pi\)
−0.552518 + 0.833501i \(0.686333\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 6311.33 1.42970 0.714852 − 0.699276i \(-0.246494\pi\)
0.714852 + 0.699276i \(0.246494\pi\)
\(12\) 0 0
\(13\) −11909.0 −1.50340 −0.751699 − 0.659506i \(-0.770765\pi\)
−0.751699 + 0.659506i \(0.770765\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 288.453 0.0142398 0.00711991 − 0.999975i \(-0.497734\pi\)
0.00711991 + 0.999975i \(0.497734\pi\)
\(18\) 0 0
\(19\) 28007.7 0.936785 0.468392 − 0.883521i \(-0.344833\pi\)
0.468392 + 0.883521i \(0.344833\pi\)
\(20\) 0 0
\(21\) −27075.9 −0.637993
\(22\) 0 0
\(23\) −28288.8 −0.484805 −0.242402 − 0.970176i \(-0.577935\pi\)
−0.242402 + 0.970176i \(0.577935\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) 97664.9 0.743610 0.371805 − 0.928311i \(-0.378739\pi\)
0.371805 + 0.928311i \(0.378739\pi\)
\(30\) 0 0
\(31\) −290434. −1.75098 −0.875491 − 0.483234i \(-0.839462\pi\)
−0.875491 + 0.483234i \(0.839462\pi\)
\(32\) 0 0
\(33\) 170406. 0.825440
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 24738.0 0.0802895 0.0401448 − 0.999194i \(-0.487218\pi\)
0.0401448 + 0.999194i \(0.487218\pi\)
\(38\) 0 0
\(39\) −321543. −0.867987
\(40\) 0 0
\(41\) 688629. 1.56042 0.780210 − 0.625517i \(-0.215112\pi\)
0.780210 + 0.625517i \(0.215112\pi\)
\(42\) 0 0
\(43\) −396038. −0.759621 −0.379811 − 0.925064i \(-0.624011\pi\)
−0.379811 + 0.925064i \(0.624011\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 920149. 1.29275 0.646377 − 0.763018i \(-0.276283\pi\)
0.646377 + 0.763018i \(0.276283\pi\)
\(48\) 0 0
\(49\) 182089. 0.221104
\(50\) 0 0
\(51\) 7788.24 0.00822136
\(52\) 0 0
\(53\) 1.64832e6 1.52082 0.760409 − 0.649445i \(-0.224999\pi\)
0.760409 + 0.649445i \(0.224999\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 756208. 0.540853
\(58\) 0 0
\(59\) 2.34227e6 1.48475 0.742377 − 0.669983i \(-0.233699\pi\)
0.742377 + 0.669983i \(0.233699\pi\)
\(60\) 0 0
\(61\) −128675. −0.0725839 −0.0362919 − 0.999341i \(-0.511555\pi\)
−0.0362919 + 0.999341i \(0.511555\pi\)
\(62\) 0 0
\(63\) −731050. −0.368345
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.69575e6 1.09501 0.547505 − 0.836802i \(-0.315578\pi\)
0.547505 + 0.836802i \(0.315578\pi\)
\(68\) 0 0
\(69\) −763797. −0.279902
\(70\) 0 0
\(71\) −2.06763e6 −0.685596 −0.342798 − 0.939409i \(-0.611375\pi\)
−0.342798 + 0.939409i \(0.611375\pi\)
\(72\) 0 0
\(73\) −2.11158e6 −0.635299 −0.317650 − 0.948208i \(-0.602894\pi\)
−0.317650 + 0.948208i \(0.602894\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.32907e6 −1.57987
\(78\) 0 0
\(79\) −694941. −0.158582 −0.0792909 − 0.996852i \(-0.525266\pi\)
−0.0792909 + 0.996852i \(0.525266\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) −4.03690e6 −0.774953 −0.387476 − 0.921880i \(-0.626653\pi\)
−0.387476 + 0.921880i \(0.626653\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.63695e6 0.429324
\(88\) 0 0
\(89\) 3.43798e6 0.516938 0.258469 − 0.966020i \(-0.416782\pi\)
0.258469 + 0.966020i \(0.416782\pi\)
\(90\) 0 0
\(91\) 1.19425e7 1.66131
\(92\) 0 0
\(93\) −7.84172e6 −1.01093
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.90999e6 0.879984 0.439992 − 0.898002i \(-0.354981\pi\)
0.439992 + 0.898002i \(0.354981\pi\)
\(98\) 0 0
\(99\) 4.60096e6 0.476568
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.n.1.1 yes 3
5.2 odd 4 300.8.d.h.49.1 6
5.3 odd 4 300.8.d.h.49.6 6
5.4 even 2 300.8.a.m.1.3 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.8.a.m.1.3 ✓ 3 5.4 even 2
300.8.a.n.1.1 yes 3 1.1 even 1 trivial
300.8.d.h.49.1 6 5.2 odd 4
300.8.d.h.49.6 6 5.3 odd 4