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 = [10,0,0,6,0,0,2,0,0,0,0,0,-14] 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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 13x^{8} + 53x^{6} - 84x^{4} + 45x^{2} - 4 \) 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 1.2
Root \(-1.67209\) 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)
 
\(f(q)\) \(=\) \(q-1.67209 q^{2} +0.795879 q^{4} -5.24127 q^{7} +2.01340 q^{8} +3.01785 q^{11} -3.80484 q^{13} +8.76386 q^{14} -4.95833 q^{16} +4.30149 q^{17} +2.28293 q^{19} -5.04610 q^{22} +2.78066 q^{23} +6.36202 q^{26} -4.17141 q^{28} -1.29327 q^{29} +6.11366 q^{31} +4.26398 q^{32} -7.19247 q^{34} -1.00000 q^{37} -3.81727 q^{38} +1.76636 q^{41} -0.682218 q^{43} +2.40184 q^{44} -4.64951 q^{46} -11.3408 q^{47} +20.4709 q^{49} -3.02819 q^{52} +5.45184 q^{53} -10.5528 q^{56} +2.16246 q^{58} +7.78323 q^{59} -5.55905 q^{61} -10.2226 q^{62} +2.78692 q^{64} -11.4596 q^{67} +3.42347 q^{68} -12.6244 q^{71} -5.51524 q^{73} +1.67209 q^{74} +1.81694 q^{76} -15.8173 q^{77} -5.43374 q^{79} -2.95350 q^{82} +8.61260 q^{83} +1.14073 q^{86} +6.07612 q^{88} +9.28024 q^{89} +19.9422 q^{91} +2.21307 q^{92} +18.9628 q^{94} +4.53086 q^{97} -34.2291 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 6 q^{4} + 2 q^{7} - 14 q^{13} + 10 q^{16} + 28 q^{19} + 28 q^{22} + 38 q^{28} + 28 q^{31} - 42 q^{34} - 10 q^{37} - 38 q^{43} + 4 q^{46} + 64 q^{49} + 4 q^{52} - 36 q^{58} + 30 q^{61} + 48 q^{64}+ \cdots - 6 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.67209 −1.18234 −0.591172 − 0.806545i \(-0.701335\pi\)
−0.591172 + 0.806545i \(0.701335\pi\)
\(3\) 0 0
\(4\) 0.795879 0.397939
\(5\) 0 0
\(6\) 0 0
\(7\) −5.24127 −1.98101 −0.990506 − 0.137466i \(-0.956104\pi\)
−0.990506 + 0.137466i \(0.956104\pi\)
\(8\) 2.01340 0.711843
\(9\) 0 0
\(10\) 0 0
\(11\) 3.01785 0.909915 0.454957 − 0.890513i \(-0.349654\pi\)
0.454957 + 0.890513i \(0.349654\pi\)
\(12\) 0 0
\(13\) −3.80484 −1.05527 −0.527636 − 0.849471i \(-0.676921\pi\)
−0.527636 + 0.849471i \(0.676921\pi\)
\(14\) 8.76386 2.34224
\(15\) 0 0
\(16\) −4.95833 −1.23958
\(17\) 4.30149 1.04326 0.521632 − 0.853170i \(-0.325323\pi\)
0.521632 + 0.853170i \(0.325323\pi\)
\(18\) 0 0
\(19\) 2.28293 0.523741 0.261870 − 0.965103i \(-0.415661\pi\)
0.261870 + 0.965103i \(0.415661\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.04610 −1.07583
\(23\) 2.78066 0.579808 0.289904 − 0.957056i \(-0.406377\pi\)
0.289904 + 0.957056i \(0.406377\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 6.36202 1.24770
\(27\) 0 0
\(28\) −4.17141 −0.788323
\(29\) −1.29327 −0.240154 −0.120077 − 0.992765i \(-0.538314\pi\)
−0.120077 + 0.992765i \(0.538314\pi\)
\(30\) 0 0
\(31\) 6.11366 1.09805 0.549023 − 0.835807i \(-0.315000\pi\)
0.549023 + 0.835807i \(0.315000\pi\)
\(32\) 4.26398 0.753772
\(33\) 0 0
\(34\) −7.19247 −1.23350
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −3.81727 −0.619242
\(39\) 0 0
\(40\) 0 0
\(41\) 1.76636 0.275858 0.137929 − 0.990442i \(-0.455955\pi\)
0.137929 + 0.990442i \(0.455955\pi\)
\(42\) 0 0
\(43\) −0.682218 −0.104037 −0.0520186 − 0.998646i \(-0.516566\pi\)
−0.0520186 + 0.998646i \(0.516566\pi\)
\(44\) 2.40184 0.362091
\(45\) 0 0
\(46\) −4.64951 −0.685533
\(47\) −11.3408 −1.65422 −0.827111 − 0.562039i \(-0.810017\pi\)
−0.827111 + 0.562039i \(0.810017\pi\)
\(48\) 0 0
\(49\) 20.4709 2.92441
\(50\) 0 0
\(51\) 0 0
\(52\) −3.02819 −0.419934
\(53\) 5.45184 0.748868 0.374434 − 0.927254i \(-0.377837\pi\)
0.374434 + 0.927254i \(0.377837\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −10.5528 −1.41017
\(57\) 0 0
\(58\) 2.16246 0.283944
\(59\) 7.78323 1.01329 0.506645 − 0.862155i \(-0.330885\pi\)
0.506645 + 0.862155i \(0.330885\pi\)
\(60\) 0 0
\(61\) −5.55905 −0.711763 −0.355882 − 0.934531i \(-0.615819\pi\)
−0.355882 + 0.934531i \(0.615819\pi\)
\(62\) −10.2226 −1.29827
\(63\) 0 0
\(64\) 2.78692 0.348365
\(65\) 0 0
\(66\) 0 0
\(67\) −11.4596 −1.40002 −0.700008 − 0.714135i \(-0.746820\pi\)
−0.700008 + 0.714135i \(0.746820\pi\)
\(68\) 3.42347 0.415156
\(69\) 0 0
\(70\) 0 0
\(71\) −12.6244 −1.49824 −0.749121 − 0.662433i \(-0.769524\pi\)
−0.749121 + 0.662433i \(0.769524\pi\)
\(72\) 0 0
\(73\) −5.51524 −0.645510 −0.322755 − 0.946482i \(-0.604609\pi\)
−0.322755 + 0.946482i \(0.604609\pi\)
\(74\) 1.67209 0.194376
\(75\) 0 0
\(76\) 1.81694 0.208417
\(77\) −15.8173 −1.80255
\(78\) 0 0
\(79\) −5.43374 −0.611344 −0.305672 − 0.952137i \(-0.598881\pi\)
−0.305672 + 0.952137i \(0.598881\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −2.95350 −0.326160
\(83\) 8.61260 0.945356 0.472678 − 0.881235i \(-0.343287\pi\)
0.472678 + 0.881235i \(0.343287\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.14073 0.123008
\(87\) 0 0
\(88\) 6.07612 0.647717
\(89\) 9.28024 0.983704 0.491852 − 0.870679i \(-0.336320\pi\)
0.491852 + 0.870679i \(0.336320\pi\)
\(90\) 0 0
\(91\) 19.9422 2.09051
\(92\) 2.21307 0.230728
\(93\) 0 0
\(94\) 18.9628 1.95586
\(95\) 0 0
\(96\) 0 0
\(97\) 4.53086 0.460039 0.230020 − 0.973186i \(-0.426121\pi\)
0.230020 + 0.973186i \(0.426121\pi\)
\(98\) −34.2291 −3.45766
\(99\) 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.ct.1.2 yes 10
3.2 odd 2 inner 8325.2.a.ct.1.9 yes 10
5.4 even 2 8325.2.a.cs.1.9 yes 10
15.14 odd 2 8325.2.a.cs.1.2 ✓ 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8325.2.a.cs.1.2 ✓ 10 15.14 odd 2
8325.2.a.cs.1.9 yes 10 5.4 even 2
8325.2.a.ct.1.2 yes 10 1.1 even 1 trivial
8325.2.a.ct.1.9 yes 10 3.2 odd 2 inner