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 1.3
Root \(27.3028\) 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)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} +1735.17 q^{7} +729.000 q^{9} +5348.61 q^{11} +8219.55 q^{13} +15753.6 q^{17} -11286.7 q^{19} +46849.5 q^{21} +85743.6 q^{23} +19683.0 q^{27} -131363. q^{29} -30750.0 q^{31} +144413. q^{33} +68121.8 q^{37} +221928. q^{39} -636968. q^{41} -691748. q^{43} +876889. q^{47} +2.18726e6 q^{49} +425348. q^{51} -1.28913e6 q^{53} -304741. q^{57} -2.52448e6 q^{59} -166980. q^{61} +1.26494e6 q^{63} -975427. q^{67} +2.31508e6 q^{69} +1.49015e6 q^{71} +6.62416e6 q^{73} +9.28074e6 q^{77} -1.95150e6 q^{79} +531441. q^{81} -6.58114e6 q^{83} -3.54679e6 q^{87} -1.99763e6 q^{89} +1.42623e7 q^{91} -830250. q^{93} +7.87194e6 q^{97} +3.89914e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 81 q^{3} + 351 q^{7} + 2187 q^{9} + 3138 q^{11} + 3585 q^{13} + 5226 q^{17} + 21195 q^{19} + 9477 q^{21} + 40434 q^{23} + 59049 q^{27} + 26598 q^{29} - 143463 q^{31} + 84726 q^{33} + 536658 q^{37}+ \cdots + 2287602 q^{99}+O(q^{100}) \) 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
\(7\) 1735.17 1.91204 0.956022 − 0.293294i \(-0.0947513\pi\)
0.956022 + 0.293294i \(0.0947513\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 5348.61 1.21162 0.605811 − 0.795609i \(-0.292849\pi\)
0.605811 + 0.795609i \(0.292849\pi\)
\(12\) 0 0
\(13\) 8219.55 1.03764 0.518820 − 0.854884i \(-0.326372\pi\)
0.518820 + 0.854884i \(0.326372\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 15753.6 0.777694 0.388847 − 0.921302i \(-0.372873\pi\)
0.388847 + 0.921302i \(0.372873\pi\)
\(18\) 0 0
\(19\) −11286.7 −0.377511 −0.188756 − 0.982024i \(-0.560445\pi\)
−0.188756 + 0.982024i \(0.560445\pi\)
\(20\) 0 0
\(21\) 46849.5 1.10392
\(22\) 0 0
\(23\) 85743.6 1.46945 0.734724 − 0.678366i \(-0.237312\pi\)
0.734724 + 0.678366i \(0.237312\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) −131363. −1.00018 −0.500091 − 0.865973i \(-0.666700\pi\)
−0.500091 + 0.865973i \(0.666700\pi\)
\(30\) 0 0
\(31\) −30750.0 −0.185387 −0.0926935 − 0.995695i \(-0.529548\pi\)
−0.0926935 + 0.995695i \(0.529548\pi\)
\(32\) 0 0
\(33\) 144413. 0.699530
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 68121.8 0.221096 0.110548 − 0.993871i \(-0.464739\pi\)
0.110548 + 0.993871i \(0.464739\pi\)
\(38\) 0 0
\(39\) 221928. 0.599081
\(40\) 0 0
\(41\) −636968. −1.44336 −0.721679 − 0.692227i \(-0.756630\pi\)
−0.721679 + 0.692227i \(0.756630\pi\)
\(42\) 0 0
\(43\) −691748. −1.32681 −0.663405 − 0.748261i \(-0.730889\pi\)
−0.663405 + 0.748261i \(0.730889\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 876889. 1.23198 0.615988 − 0.787756i \(-0.288757\pi\)
0.615988 + 0.787756i \(0.288757\pi\)
\(48\) 0 0
\(49\) 2.18726e6 2.65592
\(50\) 0 0
\(51\) 425348. 0.449002
\(52\) 0 0
\(53\) −1.28913e6 −1.18941 −0.594704 − 0.803945i \(-0.702731\pi\)
−0.594704 + 0.803945i \(0.702731\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −304741. −0.217956
\(58\) 0 0
\(59\) −2.52448e6 −1.60026 −0.800129 − 0.599828i \(-0.795236\pi\)
−0.800129 + 0.599828i \(0.795236\pi\)
\(60\) 0 0
\(61\) −166980. −0.0941913 −0.0470956 − 0.998890i \(-0.514997\pi\)
−0.0470956 + 0.998890i \(0.514997\pi\)
\(62\) 0 0
\(63\) 1.26494e6 0.637348
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −975427. −0.396217 −0.198108 − 0.980180i \(-0.563480\pi\)
−0.198108 + 0.980180i \(0.563480\pi\)
\(68\) 0 0
\(69\) 2.31508e6 0.848386
\(70\) 0 0
\(71\) 1.49015e6 0.494111 0.247056 − 0.969001i \(-0.420537\pi\)
0.247056 + 0.969001i \(0.420537\pi\)
\(72\) 0 0
\(73\) 6.62416e6 1.99297 0.996485 − 0.0837758i \(-0.0266980\pi\)
0.996485 + 0.0837758i \(0.0266980\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9.28074e6 2.31667
\(78\) 0 0
\(79\) −1.95150e6 −0.445321 −0.222660 − 0.974896i \(-0.571474\pi\)
−0.222660 + 0.974896i \(0.571474\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) −6.58114e6 −1.26336 −0.631681 − 0.775228i \(-0.717635\pi\)
−0.631681 + 0.775228i \(0.717635\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −3.54679e6 −0.577455
\(88\) 0 0
\(89\) −1.99763e6 −0.300366 −0.150183 − 0.988658i \(-0.547986\pi\)
−0.150183 + 0.988658i \(0.547986\pi\)
\(90\) 0 0
\(91\) 1.42623e7 1.98401
\(92\) 0 0
\(93\) −830250. −0.107033
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.87194e6 0.875751 0.437875 − 0.899036i \(-0.355731\pi\)
0.437875 + 0.899036i \(0.355731\pi\)
\(98\) 0 0
\(99\) 3.89914e6 0.403874
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.1.3 yes 3
5.2 odd 4 300.8.d.h.49.3 6
5.3 odd 4 300.8.d.h.49.4 6
5.4 even 2 300.8.a.m.1.1 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.8.a.m.1.1 ✓ 3 5.4 even 2
300.8.a.n.1.3 yes 3 1.1 even 1 trivial
300.8.d.h.49.3 6 5.2 odd 4
300.8.d.h.49.4 6 5.3 odd 4