Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4275,2,Mod(1,4275)] 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("4275.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,8,0,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(34.1360468641\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.16717036.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 26x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.13194\) of defining polynomial
Character \(\chi\) \(=\) 4275.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.13194 q^{2} -0.718710 q^{4} -4.11009 q^{7} +3.07742 q^{8} +5.78432 q^{11} +6.78276 q^{13} +4.65238 q^{14} -2.04604 q^{16} +5.41381 q^{17} -1.00000 q^{19} -6.54751 q^{22} +8.04820 q^{23} -7.67769 q^{26} +2.95396 q^{28} -5.34130 q^{29} +0.327327 q^{31} -3.83884 q^{32} -6.12811 q^{34} +10.7828 q^{37} +1.13194 q^{38} +2.70690 q^{41} +0.654655 q^{43} -4.15725 q^{44} -9.11009 q^{46} -7.67769 q^{47} +9.89286 q^{49} -4.87484 q^{52} -0.813537 q^{53} -12.6485 q^{56} +6.04604 q^{58} -4.97079 q^{59} -7.87484 q^{61} -0.370515 q^{62} +8.43742 q^{64} +9.76475 q^{67} -3.89096 q^{68} +4.97079 q^{71} -12.7857 q^{73} -12.2055 q^{74} +0.718710 q^{76} -23.7741 q^{77} -1.01802 q^{79} -3.06406 q^{82} -1.25656 q^{83} -0.741030 q^{86} +17.8008 q^{88} -11.4961 q^{89} -27.8778 q^{91} -5.78432 q^{92} +8.69069 q^{94} +6.78276 q^{97} -11.1981 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4} + 16 q^{13} - 6 q^{19} + 10 q^{22} + 30 q^{28} + 2 q^{31} - 12 q^{34} + 40 q^{37} + 4 q^{43} - 30 q^{46} + 10 q^{49} + 20 q^{52} + 24 q^{58} + 2 q^{61} + 26 q^{64} + 34 q^{67} + 22 q^{73}+ \cdots + 16 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.13194 −0.800403 −0.400202 − 0.916427i \(-0.631060\pi\)
−0.400202 + 0.916427i \(0.631060\pi\)
\(3\) 0 0
\(4\) −0.718710 −0.359355
\(5\) 0 0
\(6\) 0 0
\(7\) −4.11009 −1.55347 −0.776734 − 0.629828i \(-0.783125\pi\)
−0.776734 + 0.629828i \(0.783125\pi\)
\(8\) 3.07742 1.08803
\(9\) 0 0
\(10\) 0 0
\(11\) 5.78432 1.74404 0.872019 − 0.489471i \(-0.162810\pi\)
0.872019 + 0.489471i \(0.162810\pi\)
\(12\) 0 0
\(13\) 6.78276 1.88120 0.940600 − 0.339516i \(-0.110263\pi\)
0.940600 + 0.339516i \(0.110263\pi\)
\(14\) 4.65238 1.24340
\(15\) 0 0
\(16\) −2.04604 −0.511509
\(17\) 5.41381 1.31304 0.656521 − 0.754308i \(-0.272028\pi\)
0.656521 + 0.754308i \(0.272028\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −6.54751 −1.39593
\(23\) 8.04820 1.67817 0.839083 − 0.544003i \(-0.183092\pi\)
0.839083 + 0.544003i \(0.183092\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −7.67769 −1.50572
\(27\) 0 0
\(28\) 2.95396 0.558247
\(29\) −5.34130 −0.991855 −0.495927 − 0.868364i \(-0.665172\pi\)
−0.495927 + 0.868364i \(0.665172\pi\)
\(30\) 0 0
\(31\) 0.327327 0.0587897 0.0293949 − 0.999568i \(-0.490642\pi\)
0.0293949 + 0.999568i \(0.490642\pi\)
\(32\) −3.83884 −0.678618
\(33\) 0 0
\(34\) −6.12811 −1.05096
\(35\) 0 0
\(36\) 0 0
\(37\) 10.7828 1.77268 0.886338 − 0.463039i \(-0.153241\pi\)
0.886338 + 0.463039i \(0.153241\pi\)
\(38\) 1.13194 0.183625
\(39\) 0 0
\(40\) 0 0
\(41\) 2.70690 0.422747 0.211374 − 0.977405i \(-0.432206\pi\)
0.211374 + 0.977405i \(0.432206\pi\)
\(42\) 0 0
\(43\) 0.654655 0.0998339 0.0499169 − 0.998753i \(-0.484104\pi\)
0.0499169 + 0.998753i \(0.484104\pi\)
\(44\) −4.15725 −0.626729
\(45\) 0 0
\(46\) −9.11009 −1.34321
\(47\) −7.67769 −1.11991 −0.559953 − 0.828524i \(-0.689181\pi\)
−0.559953 + 0.828524i \(0.689181\pi\)
\(48\) 0 0
\(49\) 9.89286 1.41327
\(50\) 0 0
\(51\) 0 0
\(52\) −4.87484 −0.676019
\(53\) −0.813537 −0.111748 −0.0558739 − 0.998438i \(-0.517794\pi\)
−0.0558739 + 0.998438i \(0.517794\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −12.6485 −1.69022
\(57\) 0 0
\(58\) 6.04604 0.793884
\(59\) −4.97079 −0.647141 −0.323571 − 0.946204i \(-0.604883\pi\)
−0.323571 + 0.946204i \(0.604883\pi\)
\(60\) 0 0
\(61\) −7.87484 −1.00827 −0.504135 − 0.863625i \(-0.668189\pi\)
−0.504135 + 0.863625i \(0.668189\pi\)
\(62\) −0.370515 −0.0470555
\(63\) 0 0
\(64\) 8.43742 1.05468
\(65\) 0 0
\(66\) 0 0
\(67\) 9.76475 1.19295 0.596477 − 0.802630i \(-0.296567\pi\)
0.596477 + 0.802630i \(0.296567\pi\)
\(68\) −3.89096 −0.471848
\(69\) 0 0
\(70\) 0 0
\(71\) 4.97079 0.589924 0.294962 − 0.955509i \(-0.404693\pi\)
0.294962 + 0.955509i \(0.404693\pi\)
\(72\) 0 0
\(73\) −12.7857 −1.49645 −0.748227 − 0.663442i \(-0.769095\pi\)
−0.748227 + 0.663442i \(0.769095\pi\)
\(74\) −12.2055 −1.41886
\(75\) 0 0
\(76\) 0.718710 0.0824417
\(77\) −23.7741 −2.70931
\(78\) 0 0
\(79\) −1.01802 −0.114536 −0.0572680 − 0.998359i \(-0.518239\pi\)
−0.0572680 + 0.998359i \(0.518239\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −3.06406 −0.338368
\(83\) −1.25656 −0.137925 −0.0689626 − 0.997619i \(-0.521969\pi\)
−0.0689626 + 0.997619i \(0.521969\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.741030 −0.0799074
\(87\) 0 0
\(88\) 17.8008 1.89757
\(89\) −11.4961 −1.21859 −0.609294 − 0.792944i \(-0.708547\pi\)
−0.609294 + 0.792944i \(0.708547\pi\)
\(90\) 0 0
\(91\) −27.8778 −2.92239
\(92\) −5.78432 −0.603057
\(93\) 0 0
\(94\) 8.69069 0.896376
\(95\) 0 0
\(96\) 0 0
\(97\) 6.78276 0.688685 0.344343 − 0.938844i \(-0.388102\pi\)
0.344343 + 0.938844i \(0.388102\pi\)
\(98\) −11.1981 −1.13118
\(99\) 0 0
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 4275.2.a.bt.1.3 yes 6
3.2 odd 2 inner 4275.2.a.bt.1.4 yes 6
5.4 even 2 4275.2.a.bs.1.4 yes 6
15.14 odd 2 4275.2.a.bs.1.3 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4275.2.a.bs.1.3 ✓ 6 15.14 odd 2
4275.2.a.bs.1.4 yes 6 5.4 even 2
4275.2.a.bt.1.3 yes 6 1.1 even 1 trivial
4275.2.a.bt.1.4 yes 6 3.2 odd 2 inner