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 = [13,0,0,16,0,0,8,0,0,0,-10,0,11] 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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 21x^{11} + 165x^{9} - 2x^{8} - 600x^{7} + 8x^{6} + 1005x^{5} + 32x^{4} - 657x^{3} - 80x^{2} + 73x - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 555)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(1.59785\) 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)
 
\(f(q)\) \(=\) \(q+1.59785 q^{2} +0.553137 q^{4} +4.47204 q^{7} -2.31188 q^{8} +5.92804 q^{11} -5.46833 q^{13} +7.14566 q^{14} -4.80031 q^{16} -0.704359 q^{17} +2.56366 q^{19} +9.47214 q^{22} +4.43915 q^{23} -8.73760 q^{26} +2.47365 q^{28} -2.94587 q^{29} +2.35803 q^{31} -3.04645 q^{32} -1.12546 q^{34} +1.00000 q^{37} +4.09635 q^{38} -7.51337 q^{41} +12.2007 q^{43} +3.27902 q^{44} +7.09311 q^{46} +4.84149 q^{47} +12.9991 q^{49} -3.02474 q^{52} -0.897537 q^{53} -10.3388 q^{56} -4.70707 q^{58} -5.55910 q^{59} +9.19828 q^{61} +3.76779 q^{62} +4.73285 q^{64} +8.04422 q^{67} -0.389607 q^{68} -11.0300 q^{71} -3.16378 q^{73} +1.59785 q^{74} +1.41806 q^{76} +26.5104 q^{77} +6.83580 q^{79} -12.0053 q^{82} +11.3658 q^{83} +19.4949 q^{86} -13.7049 q^{88} +1.13655 q^{89} -24.4546 q^{91} +2.45546 q^{92} +7.73599 q^{94} +14.1055 q^{97} +20.7707 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 16 q^{4} + 8 q^{7} - 10 q^{11} + 11 q^{13} - 8 q^{14} + 22 q^{16} - 2 q^{17} + 14 q^{19} + 14 q^{22} + 2 q^{23} - 16 q^{26} + 4 q^{28} - 23 q^{29} + 16 q^{31} + 10 q^{32} - 10 q^{34} + 13 q^{37}+ \cdots + 42 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.59785 1.12985 0.564927 − 0.825141i \(-0.308904\pi\)
0.564927 + 0.825141i \(0.308904\pi\)
\(3\) 0 0
\(4\) 0.553137 0.276569
\(5\) 0 0
\(6\) 0 0
\(7\) 4.47204 1.69027 0.845136 − 0.534552i \(-0.179520\pi\)
0.845136 + 0.534552i \(0.179520\pi\)
\(8\) −2.31188 −0.817371
\(9\) 0 0
\(10\) 0 0
\(11\) 5.92804 1.78737 0.893686 − 0.448693i \(-0.148110\pi\)
0.893686 + 0.448693i \(0.148110\pi\)
\(12\) 0 0
\(13\) −5.46833 −1.51664 −0.758321 − 0.651881i \(-0.773980\pi\)
−0.758321 + 0.651881i \(0.773980\pi\)
\(14\) 7.14566 1.90976
\(15\) 0 0
\(16\) −4.80031 −1.20008
\(17\) −0.704359 −0.170832 −0.0854161 − 0.996345i \(-0.527222\pi\)
−0.0854161 + 0.996345i \(0.527222\pi\)
\(18\) 0 0
\(19\) 2.56366 0.588144 0.294072 − 0.955783i \(-0.404989\pi\)
0.294072 + 0.955783i \(0.404989\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 9.47214 2.01947
\(23\) 4.43915 0.925627 0.462813 − 0.886456i \(-0.346840\pi\)
0.462813 + 0.886456i \(0.346840\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −8.73760 −1.71358
\(27\) 0 0
\(28\) 2.47365 0.467476
\(29\) −2.94587 −0.547034 −0.273517 − 0.961867i \(-0.588187\pi\)
−0.273517 + 0.961867i \(0.588187\pi\)
\(30\) 0 0
\(31\) 2.35803 0.423515 0.211758 − 0.977322i \(-0.432081\pi\)
0.211758 + 0.977322i \(0.432081\pi\)
\(32\) −3.04645 −0.538541
\(33\) 0 0
\(34\) −1.12546 −0.193015
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 4.09635 0.664517
\(39\) 0 0
\(40\) 0 0
\(41\) −7.51337 −1.17339 −0.586695 − 0.809808i \(-0.699571\pi\)
−0.586695 + 0.809808i \(0.699571\pi\)
\(42\) 0 0
\(43\) 12.2007 1.86058 0.930292 − 0.366820i \(-0.119554\pi\)
0.930292 + 0.366820i \(0.119554\pi\)
\(44\) 3.27902 0.494331
\(45\) 0 0
\(46\) 7.09311 1.04582
\(47\) 4.84149 0.706203 0.353102 − 0.935585i \(-0.385127\pi\)
0.353102 + 0.935585i \(0.385127\pi\)
\(48\) 0 0
\(49\) 12.9991 1.85702
\(50\) 0 0
\(51\) 0 0
\(52\) −3.02474 −0.419456
\(53\) −0.897537 −0.123286 −0.0616431 − 0.998098i \(-0.519634\pi\)
−0.0616431 + 0.998098i \(0.519634\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −10.3388 −1.38158
\(57\) 0 0
\(58\) −4.70707 −0.618068
\(59\) −5.55910 −0.723733 −0.361866 − 0.932230i \(-0.617860\pi\)
−0.361866 + 0.932230i \(0.617860\pi\)
\(60\) 0 0
\(61\) 9.19828 1.17772 0.588860 − 0.808235i \(-0.299577\pi\)
0.588860 + 0.808235i \(0.299577\pi\)
\(62\) 3.76779 0.478510
\(63\) 0 0
\(64\) 4.73285 0.591606
\(65\) 0 0
\(66\) 0 0
\(67\) 8.04422 0.982758 0.491379 − 0.870946i \(-0.336493\pi\)
0.491379 + 0.870946i \(0.336493\pi\)
\(68\) −0.389607 −0.0472468
\(69\) 0 0
\(70\) 0 0
\(71\) −11.0300 −1.30902 −0.654510 − 0.756053i \(-0.727125\pi\)
−0.654510 + 0.756053i \(0.727125\pi\)
\(72\) 0 0
\(73\) −3.16378 −0.370293 −0.185146 − 0.982711i \(-0.559276\pi\)
−0.185146 + 0.982711i \(0.559276\pi\)
\(74\) 1.59785 0.185747
\(75\) 0 0
\(76\) 1.41806 0.162662
\(77\) 26.5104 3.02114
\(78\) 0 0
\(79\) 6.83580 0.769088 0.384544 − 0.923107i \(-0.374359\pi\)
0.384544 + 0.923107i \(0.374359\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −12.0053 −1.32576
\(83\) 11.3658 1.24756 0.623781 − 0.781599i \(-0.285596\pi\)
0.623781 + 0.781599i \(0.285596\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 19.4949 2.10219
\(87\) 0 0
\(88\) −13.7049 −1.46095
\(89\) 1.13655 0.120474 0.0602372 − 0.998184i \(-0.480814\pi\)
0.0602372 + 0.998184i \(0.480814\pi\)
\(90\) 0 0
\(91\) −24.4546 −2.56354
\(92\) 2.45546 0.255999
\(93\) 0 0
\(94\) 7.73599 0.797906
\(95\) 0 0
\(96\) 0 0
\(97\) 14.1055 1.43220 0.716100 − 0.697997i \(-0.245925\pi\)
0.716100 + 0.697997i \(0.245925\pi\)
\(98\) 20.7707 2.09816
\(99\) 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.cv.1.10 13
3.2 odd 2 2775.2.a.bh.1.4 13
5.2 odd 4 1665.2.c.f.334.20 26
5.3 odd 4 1665.2.c.f.334.7 26
5.4 even 2 8325.2.a.cu.1.4 13
15.2 even 4 555.2.c.c.334.7 ✓ 26
15.8 even 4 555.2.c.c.334.20 yes 26
15.14 odd 2 2775.2.a.bg.1.10 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.c.c.334.7 ✓ 26 15.2 even 4
555.2.c.c.334.20 yes 26 15.8 even 4
1665.2.c.f.334.7 26 5.3 odd 4
1665.2.c.f.334.20 26 5.2 odd 4
2775.2.a.bg.1.10 13 15.14 odd 2
2775.2.a.bh.1.4 13 3.2 odd 2
8325.2.a.cu.1.4 13 5.4 even 2
8325.2.a.cv.1.10 13 1.1 even 1 trivial