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 1.8
Root \(1.49190\) 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)
 
\(f(q)\) \(=\) \(q+1.49190 q^{2} +0.225760 q^{4} +2.23264 q^{7} -2.64699 q^{8} +5.15163 q^{11} +1.45307 q^{13} +3.33087 q^{14} -4.40055 q^{16} -0.150283 q^{17} -4.63319 q^{19} +7.68571 q^{22} +1.12887 q^{23} +2.16784 q^{26} +0.504042 q^{28} +1.45774 q^{29} +6.63164 q^{31} -1.27121 q^{32} -0.224207 q^{34} -1.00000 q^{37} -6.91225 q^{38} +8.43831 q^{41} +0.405880 q^{43} +1.16303 q^{44} +1.68416 q^{46} +3.42878 q^{47} -2.01531 q^{49} +0.328046 q^{52} +4.29943 q^{53} -5.90977 q^{56} +2.17479 q^{58} -12.1777 q^{59} +0.826761 q^{61} +9.89373 q^{62} +6.90459 q^{64} -9.70947 q^{67} -0.0339279 q^{68} +8.73070 q^{71} +4.18700 q^{73} -1.49190 q^{74} -1.04599 q^{76} +11.5018 q^{77} +9.00843 q^{79} +12.5891 q^{82} +3.54362 q^{83} +0.605532 q^{86} -13.6363 q^{88} -5.74690 q^{89} +3.24419 q^{91} +0.254854 q^{92} +5.11539 q^{94} +1.50129 q^{97} -3.00664 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 6 q^{4} + 2 q^{7} - 14 q^{13} + 10 q^{16} + 28 q^{19} + 28 q^{22} + 38 q^{28} + 28 q^{31} - 42 q^{34} - 10 q^{37} - 38 q^{43} + 4 q^{46} + 64 q^{49} + 4 q^{52} - 36 q^{58} + 30 q^{61} + 48 q^{64}+ \cdots - 6 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.49190 1.05493 0.527466 − 0.849576i \(-0.323142\pi\)
0.527466 + 0.849576i \(0.323142\pi\)
\(3\) 0 0
\(4\) 0.225760 0.112880
\(5\) 0 0
\(6\) 0 0
\(7\) 2.23264 0.843859 0.421929 − 0.906629i \(-0.361353\pi\)
0.421929 + 0.906629i \(0.361353\pi\)
\(8\) −2.64699 −0.935851
\(9\) 0 0
\(10\) 0 0
\(11\) 5.15163 1.55328 0.776638 − 0.629947i \(-0.216923\pi\)
0.776638 + 0.629947i \(0.216923\pi\)
\(12\) 0 0
\(13\) 1.45307 0.403010 0.201505 − 0.979487i \(-0.435417\pi\)
0.201505 + 0.979487i \(0.435417\pi\)
\(14\) 3.33087 0.890213
\(15\) 0 0
\(16\) −4.40055 −1.10014
\(17\) −0.150283 −0.0364490 −0.0182245 − 0.999834i \(-0.505801\pi\)
−0.0182245 + 0.999834i \(0.505801\pi\)
\(18\) 0 0
\(19\) −4.63319 −1.06293 −0.531464 − 0.847081i \(-0.678358\pi\)
−0.531464 + 0.847081i \(0.678358\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 7.68571 1.63860
\(23\) 1.12887 0.235386 0.117693 − 0.993050i \(-0.462450\pi\)
0.117693 + 0.993050i \(0.462450\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.16784 0.425148
\(27\) 0 0
\(28\) 0.504042 0.0952549
\(29\) 1.45774 0.270695 0.135347 − 0.990798i \(-0.456785\pi\)
0.135347 + 0.990798i \(0.456785\pi\)
\(30\) 0 0
\(31\) 6.63164 1.19108 0.595539 − 0.803326i \(-0.296939\pi\)
0.595539 + 0.803326i \(0.296939\pi\)
\(32\) −1.27121 −0.224720
\(33\) 0 0
\(34\) −0.224207 −0.0384512
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −6.91225 −1.12132
\(39\) 0 0
\(40\) 0 0
\(41\) 8.43831 1.31784 0.658922 − 0.752212i \(-0.271013\pi\)
0.658922 + 0.752212i \(0.271013\pi\)
\(42\) 0 0
\(43\) 0.405880 0.0618961 0.0309481 − 0.999521i \(-0.490147\pi\)
0.0309481 + 0.999521i \(0.490147\pi\)
\(44\) 1.16303 0.175334
\(45\) 0 0
\(46\) 1.68416 0.248316
\(47\) 3.42878 0.500139 0.250069 − 0.968228i \(-0.419547\pi\)
0.250069 + 0.968228i \(0.419547\pi\)
\(48\) 0 0
\(49\) −2.01531 −0.287902
\(50\) 0 0
\(51\) 0 0
\(52\) 0.328046 0.0454918
\(53\) 4.29943 0.590572 0.295286 − 0.955409i \(-0.404585\pi\)
0.295286 + 0.955409i \(0.404585\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −5.90977 −0.789726
\(57\) 0 0
\(58\) 2.17479 0.285564
\(59\) −12.1777 −1.58540 −0.792700 − 0.609612i \(-0.791325\pi\)
−0.792700 + 0.609612i \(0.791325\pi\)
\(60\) 0 0
\(61\) 0.826761 0.105856 0.0529279 − 0.998598i \(-0.483145\pi\)
0.0529279 + 0.998598i \(0.483145\pi\)
\(62\) 9.89373 1.25651
\(63\) 0 0
\(64\) 6.90459 0.863074
\(65\) 0 0
\(66\) 0 0
\(67\) −9.70947 −1.18620 −0.593101 − 0.805128i \(-0.702096\pi\)
−0.593101 + 0.805128i \(0.702096\pi\)
\(68\) −0.0339279 −0.00411437
\(69\) 0 0
\(70\) 0 0
\(71\) 8.73070 1.03614 0.518072 − 0.855337i \(-0.326650\pi\)
0.518072 + 0.855337i \(0.326650\pi\)
\(72\) 0 0
\(73\) 4.18700 0.490051 0.245026 − 0.969517i \(-0.421204\pi\)
0.245026 + 0.969517i \(0.421204\pi\)
\(74\) −1.49190 −0.173430
\(75\) 0 0
\(76\) −1.04599 −0.119983
\(77\) 11.5018 1.31075
\(78\) 0 0
\(79\) 9.00843 1.01353 0.506764 − 0.862085i \(-0.330842\pi\)
0.506764 + 0.862085i \(0.330842\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 12.5891 1.39023
\(83\) 3.54362 0.388962 0.194481 − 0.980906i \(-0.437698\pi\)
0.194481 + 0.980906i \(0.437698\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.605532 0.0652962
\(87\) 0 0
\(88\) −13.6363 −1.45363
\(89\) −5.74690 −0.609170 −0.304585 − 0.952485i \(-0.598518\pi\)
−0.304585 + 0.952485i \(0.598518\pi\)
\(90\) 0 0
\(91\) 3.24419 0.340084
\(92\) 0.254854 0.0265704
\(93\) 0 0
\(94\) 5.11539 0.527612
\(95\) 0 0
\(96\) 0 0
\(97\) 1.50129 0.152432 0.0762162 − 0.997091i \(-0.475716\pi\)
0.0762162 + 0.997091i \(0.475716\pi\)
\(98\) −3.00664 −0.303717
\(99\) 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.1.8 yes 10
3.2 odd 2 inner 8325.2.a.ct.1.3 yes 10
5.4 even 2 8325.2.a.cs.1.3 ✓ 10
15.14 odd 2 8325.2.a.cs.1.8 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8325.2.a.cs.1.3 ✓ 10 5.4 even 2
8325.2.a.cs.1.8 yes 10 15.14 odd 2
8325.2.a.ct.1.3 yes 10 3.2 odd 2 inner
8325.2.a.ct.1.8 yes 10 1.1 even 1 trivial