Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-4,0,12,0,0,1,-9,0,0,5,0,1] 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: \(66.4754596827\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.297869800.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 9x^{4} + 7x^{3} + 23x^{2} + 2x - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2775)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.52727\) of defining polynomial
Character \(\chi\) \(=\) 8325.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.52727 q^{2} +4.38709 q^{4} +3.70785 q^{7} -6.03281 q^{8} +5.99544 q^{11} -0.118457 q^{13} -9.37074 q^{14} +6.47237 q^{16} -4.40604 q^{17} +4.28009 q^{19} -15.1521 q^{22} +6.92875 q^{23} +0.299372 q^{26} +16.2667 q^{28} +0.451007 q^{29} +9.98365 q^{31} -4.29178 q^{32} +11.1353 q^{34} -1.00000 q^{37} -10.8169 q^{38} -4.01439 q^{41} -3.58663 q^{43} +26.3025 q^{44} -17.5108 q^{46} +10.1045 q^{47} +6.74817 q^{49} -0.519681 q^{52} -3.29215 q^{53} -22.3688 q^{56} -1.13982 q^{58} -2.69640 q^{59} -4.97791 q^{61} -25.2314 q^{62} -2.09824 q^{64} -6.11238 q^{67} -19.3297 q^{68} +15.9308 q^{71} +8.89980 q^{73} +2.52727 q^{74} +18.7771 q^{76} +22.2302 q^{77} -2.69783 q^{79} +10.1454 q^{82} -14.1210 q^{83} +9.06437 q^{86} -36.1694 q^{88} +3.95512 q^{89} -0.439220 q^{91} +30.3970 q^{92} -25.5368 q^{94} -11.3270 q^{97} -17.0544 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{2} + 12 q^{4} + q^{7} - 9 q^{8} + 5 q^{11} + q^{13} + 4 q^{14} + 20 q^{16} - 15 q^{17} + 11 q^{19} + 10 q^{22} - 12 q^{23} + 7 q^{26} + 11 q^{28} + 9 q^{29} + 14 q^{31} - 17 q^{32} + 9 q^{34}+ \cdots - 31 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.52727 −1.78705 −0.893525 − 0.449014i \(-0.851775\pi\)
−0.893525 + 0.449014i \(0.851775\pi\)
\(3\) 0 0
\(4\) 4.38709 2.19354
\(5\) 0 0
\(6\) 0 0
\(7\) 3.70785 1.40144 0.700718 − 0.713438i \(-0.252863\pi\)
0.700718 + 0.713438i \(0.252863\pi\)
\(8\) −6.03281 −2.13292
\(9\) 0 0
\(10\) 0 0
\(11\) 5.99544 1.80769 0.903846 − 0.427857i \(-0.140731\pi\)
0.903846 + 0.427857i \(0.140731\pi\)
\(12\) 0 0
\(13\) −0.118457 −0.0328540 −0.0164270 − 0.999865i \(-0.505229\pi\)
−0.0164270 + 0.999865i \(0.505229\pi\)
\(14\) −9.37074 −2.50444
\(15\) 0 0
\(16\) 6.47237 1.61809
\(17\) −4.40604 −1.06862 −0.534311 − 0.845288i \(-0.679429\pi\)
−0.534311 + 0.845288i \(0.679429\pi\)
\(18\) 0 0
\(19\) 4.28009 0.981919 0.490960 − 0.871182i \(-0.336646\pi\)
0.490960 + 0.871182i \(0.336646\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −15.1521 −3.23043
\(23\) 6.92875 1.44474 0.722372 − 0.691505i \(-0.243052\pi\)
0.722372 + 0.691505i \(0.243052\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.299372 0.0587117
\(27\) 0 0
\(28\) 16.2667 3.07411
\(29\) 0.451007 0.0837499 0.0418750 − 0.999123i \(-0.486667\pi\)
0.0418750 + 0.999123i \(0.486667\pi\)
\(30\) 0 0
\(31\) 9.98365 1.79312 0.896558 − 0.442926i \(-0.146059\pi\)
0.896558 + 0.442926i \(0.146059\pi\)
\(32\) −4.29178 −0.758687
\(33\) 0 0
\(34\) 11.1353 1.90968
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −10.8169 −1.75474
\(39\) 0 0
\(40\) 0 0
\(41\) −4.01439 −0.626943 −0.313471 − 0.949598i \(-0.601492\pi\)
−0.313471 + 0.949598i \(0.601492\pi\)
\(42\) 0 0
\(43\) −3.58663 −0.546955 −0.273478 − 0.961878i \(-0.588174\pi\)
−0.273478 + 0.961878i \(0.588174\pi\)
\(44\) 26.3025 3.96525
\(45\) 0 0
\(46\) −17.5108 −2.58183
\(47\) 10.1045 1.47389 0.736947 − 0.675950i \(-0.236267\pi\)
0.736947 + 0.675950i \(0.236267\pi\)
\(48\) 0 0
\(49\) 6.74817 0.964024
\(50\) 0 0
\(51\) 0 0
\(52\) −0.519681 −0.0720667
\(53\) −3.29215 −0.452211 −0.226106 − 0.974103i \(-0.572599\pi\)
−0.226106 + 0.974103i \(0.572599\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −22.3688 −2.98915
\(57\) 0 0
\(58\) −1.13982 −0.149665
\(59\) −2.69640 −0.351041 −0.175521 − 0.984476i \(-0.556161\pi\)
−0.175521 + 0.984476i \(0.556161\pi\)
\(60\) 0 0
\(61\) −4.97791 −0.637356 −0.318678 − 0.947863i \(-0.603239\pi\)
−0.318678 + 0.947863i \(0.603239\pi\)
\(62\) −25.2314 −3.20439
\(63\) 0 0
\(64\) −2.09824 −0.262280
\(65\) 0 0
\(66\) 0 0
\(67\) −6.11238 −0.746746 −0.373373 − 0.927681i \(-0.621799\pi\)
−0.373373 + 0.927681i \(0.621799\pi\)
\(68\) −19.3297 −2.34407
\(69\) 0 0
\(70\) 0 0
\(71\) 15.9308 1.89064 0.945320 − 0.326143i \(-0.105749\pi\)
0.945320 + 0.326143i \(0.105749\pi\)
\(72\) 0 0
\(73\) 8.89980 1.04164 0.520821 − 0.853666i \(-0.325626\pi\)
0.520821 + 0.853666i \(0.325626\pi\)
\(74\) 2.52727 0.293789
\(75\) 0 0
\(76\) 18.7771 2.15388
\(77\) 22.2302 2.53337
\(78\) 0 0
\(79\) −2.69783 −0.303529 −0.151765 − 0.988417i \(-0.548496\pi\)
−0.151765 + 0.988417i \(0.548496\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 10.1454 1.12038
\(83\) −14.1210 −1.54998 −0.774992 − 0.631971i \(-0.782246\pi\)
−0.774992 + 0.631971i \(0.782246\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 9.06437 0.977436
\(87\) 0 0
\(88\) −36.1694 −3.85567
\(89\) 3.95512 0.419242 0.209621 − 0.977783i \(-0.432777\pi\)
0.209621 + 0.977783i \(0.432777\pi\)
\(90\) 0 0
\(91\) −0.439220 −0.0460428
\(92\) 30.3970 3.16911
\(93\) 0 0
\(94\) −25.5368 −2.63392
\(95\) 0 0
\(96\) 0 0
\(97\) −11.3270 −1.15009 −0.575043 − 0.818123i \(-0.695015\pi\)
−0.575043 + 0.818123i \(0.695015\pi\)
\(98\) −17.0544 −1.72276
\(99\) 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 8325.2.a.ci.1.2 6
3.2 odd 2 2775.2.a.bf.1.5 yes 6
5.4 even 2 8325.2.a.cl.1.5 6
15.14 odd 2 2775.2.a.be.1.2 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2775.2.a.be.1.2 ✓ 6 15.14 odd 2
2775.2.a.bf.1.5 yes 6 3.2 odd 2
8325.2.a.ci.1.2 6 1.1 even 1 trivial
8325.2.a.cl.1.5 6 5.4 even 2