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 1.10
Root \(2.65283\) 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)
 
\(f(q)\) \(=\) \(q+2.65283 q^{2} +5.03751 q^{4} -4.66099 q^{7} +8.05799 q^{8} -2.92678 q^{11} +0.896733 q^{13} -12.3648 q^{14} +11.3015 q^{16} -6.41174 q^{17} +8.64049 q^{19} -7.76425 q^{22} +5.17787 q^{23} +2.37888 q^{26} -23.4798 q^{28} +6.91299 q^{29} +5.33126 q^{31} +13.8649 q^{32} -17.0093 q^{34} +1.00000 q^{37} +22.9217 q^{38} +10.7031 q^{41} +5.70625 q^{43} -14.7437 q^{44} +13.7360 q^{46} -9.93927 q^{47} +14.7248 q^{49} +4.51730 q^{52} -5.29793 q^{53} -37.5582 q^{56} +18.3390 q^{58} +8.60998 q^{59} +9.36723 q^{61} +14.1429 q^{62} +14.1783 q^{64} +7.14231 q^{67} -32.2992 q^{68} +0.600750 q^{71} +9.12028 q^{73} +2.65283 q^{74} +43.5265 q^{76} +13.6417 q^{77} -5.34827 q^{79} +28.3935 q^{82} +3.42803 q^{83} +15.1377 q^{86} -23.5840 q^{88} -5.90641 q^{89} -4.17966 q^{91} +26.0835 q^{92} -26.3672 q^{94} +11.8845 q^{97} +39.0624 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 6 q^{4} - 2 q^{7} + 14 q^{13} + 10 q^{16} + 28 q^{19} - 28 q^{22} - 38 q^{28} + 28 q^{31} - 42 q^{34} + 10 q^{37} + 38 q^{43} + 4 q^{46} + 64 q^{49} - 4 q^{52} + 36 q^{58} + 30 q^{61} + 48 q^{64}+ \cdots + 6 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.65283 1.87583 0.937917 − 0.346859i \(-0.112752\pi\)
0.937917 + 0.346859i \(0.112752\pi\)
\(3\) 0 0
\(4\) 5.03751 2.51875
\(5\) 0 0
\(6\) 0 0
\(7\) −4.66099 −1.76169 −0.880844 − 0.473407i \(-0.843024\pi\)
−0.880844 + 0.473407i \(0.843024\pi\)
\(8\) 8.05799 2.84893
\(9\) 0 0
\(10\) 0 0
\(11\) −2.92678 −0.882458 −0.441229 − 0.897395i \(-0.645457\pi\)
−0.441229 + 0.897395i \(0.645457\pi\)
\(12\) 0 0
\(13\) 0.896733 0.248709 0.124354 − 0.992238i \(-0.460314\pi\)
0.124354 + 0.992238i \(0.460314\pi\)
\(14\) −12.3648 −3.30463
\(15\) 0 0
\(16\) 11.3015 2.82537
\(17\) −6.41174 −1.55508 −0.777538 − 0.628836i \(-0.783532\pi\)
−0.777538 + 0.628836i \(0.783532\pi\)
\(18\) 0 0
\(19\) 8.64049 1.98226 0.991132 − 0.132883i \(-0.0424233\pi\)
0.991132 + 0.132883i \(0.0424233\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −7.76425 −1.65534
\(23\) 5.17787 1.07966 0.539830 − 0.841774i \(-0.318489\pi\)
0.539830 + 0.841774i \(0.318489\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.37888 0.466537
\(27\) 0 0
\(28\) −23.4798 −4.43726
\(29\) 6.91299 1.28371 0.641855 − 0.766826i \(-0.278165\pi\)
0.641855 + 0.766826i \(0.278165\pi\)
\(30\) 0 0
\(31\) 5.33126 0.957523 0.478762 − 0.877945i \(-0.341086\pi\)
0.478762 + 0.877945i \(0.341086\pi\)
\(32\) 13.8649 2.45099
\(33\) 0 0
\(34\) −17.0093 −2.91706
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 22.9217 3.71840
\(39\) 0 0
\(40\) 0 0
\(41\) 10.7031 1.67154 0.835771 − 0.549078i \(-0.185021\pi\)
0.835771 + 0.549078i \(0.185021\pi\)
\(42\) 0 0
\(43\) 5.70625 0.870194 0.435097 − 0.900384i \(-0.356714\pi\)
0.435097 + 0.900384i \(0.356714\pi\)
\(44\) −14.7437 −2.22269
\(45\) 0 0
\(46\) 13.7360 2.02526
\(47\) −9.93927 −1.44979 −0.724896 − 0.688858i \(-0.758112\pi\)
−0.724896 + 0.688858i \(0.758112\pi\)
\(48\) 0 0
\(49\) 14.7248 2.10354
\(50\) 0 0
\(51\) 0 0
\(52\) 4.51730 0.626436
\(53\) −5.29793 −0.727727 −0.363864 − 0.931452i \(-0.618543\pi\)
−0.363864 + 0.931452i \(0.618543\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −37.5582 −5.01893
\(57\) 0 0
\(58\) 18.3390 2.40803
\(59\) 8.60998 1.12092 0.560462 − 0.828180i \(-0.310624\pi\)
0.560462 + 0.828180i \(0.310624\pi\)
\(60\) 0 0
\(61\) 9.36723 1.19935 0.599676 − 0.800243i \(-0.295296\pi\)
0.599676 + 0.800243i \(0.295296\pi\)
\(62\) 14.1429 1.79615
\(63\) 0 0
\(64\) 14.1783 1.77229
\(65\) 0 0
\(66\) 0 0
\(67\) 7.14231 0.872573 0.436286 − 0.899808i \(-0.356293\pi\)
0.436286 + 0.899808i \(0.356293\pi\)
\(68\) −32.2992 −3.91685
\(69\) 0 0
\(70\) 0 0
\(71\) 0.600750 0.0712959 0.0356480 − 0.999364i \(-0.488650\pi\)
0.0356480 + 0.999364i \(0.488650\pi\)
\(72\) 0 0
\(73\) 9.12028 1.06745 0.533724 − 0.845659i \(-0.320792\pi\)
0.533724 + 0.845659i \(0.320792\pi\)
\(74\) 2.65283 0.308385
\(75\) 0 0
\(76\) 43.5265 4.99283
\(77\) 13.6417 1.55461
\(78\) 0 0
\(79\) −5.34827 −0.601728 −0.300864 − 0.953667i \(-0.597275\pi\)
−0.300864 + 0.953667i \(0.597275\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 28.3935 3.13554
\(83\) 3.42803 0.376275 0.188138 − 0.982143i \(-0.439755\pi\)
0.188138 + 0.982143i \(0.439755\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 15.1377 1.63234
\(87\) 0 0
\(88\) −23.5840 −2.51406
\(89\) −5.90641 −0.626078 −0.313039 − 0.949740i \(-0.601347\pi\)
−0.313039 + 0.949740i \(0.601347\pi\)
\(90\) 0 0
\(91\) −4.17966 −0.438147
\(92\) 26.0835 2.71940
\(93\) 0 0
\(94\) −26.3672 −2.71957
\(95\) 0 0
\(96\) 0 0
\(97\) 11.8845 1.20669 0.603346 − 0.797480i \(-0.293834\pi\)
0.603346 + 0.797480i \(0.293834\pi\)
\(98\) 39.0624 3.94590
\(99\) 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.cs.1.10 yes 10
3.2 odd 2 inner 8325.2.a.cs.1.1 ✓ 10
5.4 even 2 8325.2.a.ct.1.1 yes 10
15.14 odd 2 8325.2.a.ct.1.10 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8325.2.a.cs.1.1 ✓ 10 3.2 odd 2 inner
8325.2.a.cs.1.10 yes 10 1.1 even 1 trivial
8325.2.a.ct.1.1 yes 10 5.4 even 2
8325.2.a.ct.1.10 yes 10 15.14 odd 2