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 = [5,-1,0,1,0,0,0,0,0,0,16,0,3] 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: \(5\)
Coefficient field: 5.5.65657.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 5x^{3} + 2x^{2} + 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 925)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.37487\) 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)
 
\(f(q)\) \(=\) \(q+1.83708 q^{2} +1.37487 q^{4} +3.54814 q^{7} -1.14842 q^{8} +1.74184 q^{11} -1.99247 q^{13} +6.51822 q^{14} -4.85947 q^{16} +6.52067 q^{17} -4.61674 q^{19} +3.19991 q^{22} -9.26795 q^{23} -3.66033 q^{26} +4.87822 q^{28} +9.40396 q^{29} +7.06881 q^{31} -6.63041 q^{32} +11.9790 q^{34} -1.00000 q^{37} -8.48132 q^{38} +2.07106 q^{41} +5.12096 q^{43} +2.39480 q^{44} -17.0260 q^{46} +8.12798 q^{47} +5.58927 q^{49} -2.73939 q^{52} +14.3177 q^{53} -4.07475 q^{56} +17.2758 q^{58} +1.14805 q^{59} -2.40658 q^{61} +12.9860 q^{62} -2.46166 q^{64} +6.16820 q^{67} +8.96506 q^{68} +5.13300 q^{71} -12.9850 q^{73} -1.83708 q^{74} -6.34740 q^{76} +6.18030 q^{77} -7.83145 q^{79} +3.80470 q^{82} -5.45121 q^{83} +9.40761 q^{86} -2.00036 q^{88} -4.14073 q^{89} -7.06956 q^{91} -12.7422 q^{92} +14.9318 q^{94} +11.4979 q^{97} +10.2680 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} + q^{4} + 16 q^{11} + 3 q^{13} + 5 q^{14} - 11 q^{16} + 2 q^{17} - 11 q^{19} - 19 q^{22} - 7 q^{23} + 4 q^{26} + 7 q^{28} + 15 q^{29} - 13 q^{31} - q^{32} + 13 q^{34} - 5 q^{37} - 2 q^{38}+ \cdots + 12 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.83708 1.29901 0.649506 − 0.760356i \(-0.274976\pi\)
0.649506 + 0.760356i \(0.274976\pi\)
\(3\) 0 0
\(4\) 1.37487 0.687434
\(5\) 0 0
\(6\) 0 0
\(7\) 3.54814 1.34107 0.670535 − 0.741878i \(-0.266065\pi\)
0.670535 + 0.741878i \(0.266065\pi\)
\(8\) −1.14842 −0.406027
\(9\) 0 0
\(10\) 0 0
\(11\) 1.74184 0.525185 0.262593 − 0.964907i \(-0.415422\pi\)
0.262593 + 0.964907i \(0.415422\pi\)
\(12\) 0 0
\(13\) −1.99247 −0.552612 −0.276306 − 0.961070i \(-0.589110\pi\)
−0.276306 + 0.961070i \(0.589110\pi\)
\(14\) 6.51822 1.74207
\(15\) 0 0
\(16\) −4.85947 −1.21487
\(17\) 6.52067 1.58150 0.790748 − 0.612142i \(-0.209692\pi\)
0.790748 + 0.612142i \(0.209692\pi\)
\(18\) 0 0
\(19\) −4.61674 −1.05915 −0.529576 − 0.848262i \(-0.677649\pi\)
−0.529576 + 0.848262i \(0.677649\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.19991 0.682222
\(23\) −9.26795 −1.93250 −0.966251 − 0.257604i \(-0.917067\pi\)
−0.966251 + 0.257604i \(0.917067\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3.66033 −0.717850
\(27\) 0 0
\(28\) 4.87822 0.921897
\(29\) 9.40396 1.74627 0.873136 − 0.487477i \(-0.162083\pi\)
0.873136 + 0.487477i \(0.162083\pi\)
\(30\) 0 0
\(31\) 7.06881 1.26960 0.634798 − 0.772678i \(-0.281083\pi\)
0.634798 + 0.772678i \(0.281083\pi\)
\(32\) −6.63041 −1.17210
\(33\) 0 0
\(34\) 11.9790 2.05438
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −8.48132 −1.37585
\(39\) 0 0
\(40\) 0 0
\(41\) 2.07106 0.323445 0.161723 − 0.986836i \(-0.448295\pi\)
0.161723 + 0.986836i \(0.448295\pi\)
\(42\) 0 0
\(43\) 5.12096 0.780938 0.390469 − 0.920616i \(-0.372313\pi\)
0.390469 + 0.920616i \(0.372313\pi\)
\(44\) 2.39480 0.361030
\(45\) 0 0
\(46\) −17.0260 −2.51034
\(47\) 8.12798 1.18559 0.592794 − 0.805354i \(-0.298025\pi\)
0.592794 + 0.805354i \(0.298025\pi\)
\(48\) 0 0
\(49\) 5.58927 0.798468
\(50\) 0 0
\(51\) 0 0
\(52\) −2.73939 −0.379884
\(53\) 14.3177 1.96668 0.983340 − 0.181774i \(-0.0581839\pi\)
0.983340 + 0.181774i \(0.0581839\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.07475 −0.544511
\(57\) 0 0
\(58\) 17.2758 2.26843
\(59\) 1.14805 0.149464 0.0747320 − 0.997204i \(-0.476190\pi\)
0.0747320 + 0.997204i \(0.476190\pi\)
\(60\) 0 0
\(61\) −2.40658 −0.308131 −0.154065 − 0.988061i \(-0.549237\pi\)
−0.154065 + 0.988061i \(0.549237\pi\)
\(62\) 12.9860 1.64922
\(63\) 0 0
\(64\) −2.46166 −0.307707
\(65\) 0 0
\(66\) 0 0
\(67\) 6.16820 0.753565 0.376783 − 0.926302i \(-0.377030\pi\)
0.376783 + 0.926302i \(0.377030\pi\)
\(68\) 8.96506 1.08717
\(69\) 0 0
\(70\) 0 0
\(71\) 5.13300 0.609175 0.304587 − 0.952484i \(-0.401481\pi\)
0.304587 + 0.952484i \(0.401481\pi\)
\(72\) 0 0
\(73\) −12.9850 −1.51978 −0.759890 − 0.650052i \(-0.774747\pi\)
−0.759890 + 0.650052i \(0.774747\pi\)
\(74\) −1.83708 −0.213556
\(75\) 0 0
\(76\) −6.34740 −0.728097
\(77\) 6.18030 0.704310
\(78\) 0 0
\(79\) −7.83145 −0.881107 −0.440554 − 0.897726i \(-0.645218\pi\)
−0.440554 + 0.897726i \(0.645218\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.80470 0.420159
\(83\) −5.45121 −0.598348 −0.299174 − 0.954199i \(-0.596711\pi\)
−0.299174 + 0.954199i \(0.596711\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 9.40761 1.01445
\(87\) 0 0
\(88\) −2.00036 −0.213240
\(89\) −4.14073 −0.438917 −0.219458 − 0.975622i \(-0.570429\pi\)
−0.219458 + 0.975622i \(0.570429\pi\)
\(90\) 0 0
\(91\) −7.06956 −0.741092
\(92\) −12.7422 −1.32847
\(93\) 0 0
\(94\) 14.9318 1.54009
\(95\) 0 0
\(96\) 0 0
\(97\) 11.4979 1.16744 0.583719 − 0.811956i \(-0.301597\pi\)
0.583719 + 0.811956i \(0.301597\pi\)
\(98\) 10.2680 1.03722
\(99\) 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.cb.1.5 5
3.2 odd 2 925.2.a.i.1.1 yes 5
5.4 even 2 8325.2.a.cd.1.1 5
15.2 even 4 925.2.b.h.149.2 10
15.8 even 4 925.2.b.h.149.9 10
15.14 odd 2 925.2.a.g.1.5 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
925.2.a.g.1.5 ✓ 5 15.14 odd 2
925.2.a.i.1.1 yes 5 3.2 odd 2
925.2.b.h.149.2 10 15.2 even 4
925.2.b.h.149.9 10 15.8 even 4
8325.2.a.cb.1.5 5 1.1 even 1 trivial
8325.2.a.cd.1.1 5 5.4 even 2