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 = [4,0,0,4,0,0,-4,-6,0,0,0,0,-4] 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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.6224.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - 2x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 111)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.44579\) 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)
 
\(f(q)\) \(=\) \(q-2.44579 q^{2} +3.98187 q^{4} -0.269264 q^{7} -4.84724 q^{8} -5.96375 q^{11} -0.658562 q^{13} +0.658562 q^{14} +3.89157 q^{16} +5.29303 q^{17} +3.07217 q^{19} +14.5861 q^{22} -9.02377 q^{23} +1.61070 q^{26} -1.07217 q^{28} +4.49012 q^{29} +1.73074 q^{31} +0.176523 q^{32} -12.9456 q^{34} +1.00000 q^{37} -7.51388 q^{38} -7.16084 q^{41} +8.05241 q^{43} -23.7469 q^{44} +22.0702 q^{46} -8.62231 q^{47} -6.92750 q^{49} -2.62231 q^{52} +5.81940 q^{53} +1.30519 q^{56} -10.9819 q^{58} +12.0646 q^{59} +7.78315 q^{61} -4.23301 q^{62} -8.21489 q^{64} +0.269264 q^{67} +21.0762 q^{68} -3.30519 q^{71} +3.69448 q^{73} -2.44579 q^{74} +12.2330 q^{76} +1.60582 q^{77} +8.85532 q^{79} +17.5139 q^{82} -5.34144 q^{83} -19.6945 q^{86} +28.9077 q^{88} -5.20925 q^{89} +0.177327 q^{91} -35.9315 q^{92} +21.0883 q^{94} -12.0475 q^{97} +16.9432 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 4 q^{7} - 6 q^{8} - 4 q^{13} + 4 q^{14} - 4 q^{16} - 2 q^{17} + 8 q^{19} + 12 q^{22} - 10 q^{23} + 8 q^{26} + 2 q^{29} + 4 q^{31} - 12 q^{32} - 16 q^{34} + 4 q^{37} + 12 q^{38} - 12 q^{41}+ \cdots + 24 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.44579 −1.72943 −0.864716 − 0.502261i \(-0.832502\pi\)
−0.864716 + 0.502261i \(0.832502\pi\)
\(3\) 0 0
\(4\) 3.98187 1.99094
\(5\) 0 0
\(6\) 0 0
\(7\) −0.269264 −0.101772 −0.0508861 − 0.998704i \(-0.516205\pi\)
−0.0508861 + 0.998704i \(0.516205\pi\)
\(8\) −4.84724 −1.71376
\(9\) 0 0
\(10\) 0 0
\(11\) −5.96375 −1.79814 −0.899069 − 0.437807i \(-0.855755\pi\)
−0.899069 + 0.437807i \(0.855755\pi\)
\(12\) 0 0
\(13\) −0.658562 −0.182652 −0.0913261 − 0.995821i \(-0.529111\pi\)
−0.0913261 + 0.995821i \(0.529111\pi\)
\(14\) 0.658562 0.176008
\(15\) 0 0
\(16\) 3.89157 0.972894
\(17\) 5.29303 1.28375 0.641874 − 0.766810i \(-0.278157\pi\)
0.641874 + 0.766810i \(0.278157\pi\)
\(18\) 0 0
\(19\) 3.07217 0.704805 0.352403 − 0.935849i \(-0.385365\pi\)
0.352403 + 0.935849i \(0.385365\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 14.5861 3.10976
\(23\) −9.02377 −1.88159 −0.940793 − 0.338983i \(-0.889917\pi\)
−0.940793 + 0.338983i \(0.889917\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.61070 0.315885
\(27\) 0 0
\(28\) −1.07217 −0.202622
\(29\) 4.49012 0.833794 0.416897 − 0.908954i \(-0.363118\pi\)
0.416897 + 0.908954i \(0.363118\pi\)
\(30\) 0 0
\(31\) 1.73074 0.310849 0.155425 − 0.987848i \(-0.450325\pi\)
0.155425 + 0.987848i \(0.450325\pi\)
\(32\) 0.176523 0.0312052
\(33\) 0 0
\(34\) −12.9456 −2.22016
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −7.51388 −1.21891
\(39\) 0 0
\(40\) 0 0
\(41\) −7.16084 −1.11833 −0.559167 − 0.829055i \(-0.688879\pi\)
−0.559167 + 0.829055i \(0.688879\pi\)
\(42\) 0 0
\(43\) 8.05241 1.22798 0.613991 − 0.789313i \(-0.289563\pi\)
0.613991 + 0.789313i \(0.289563\pi\)
\(44\) −23.7469 −3.57998
\(45\) 0 0
\(46\) 22.0702 3.25407
\(47\) −8.62231 −1.25769 −0.628847 − 0.777529i \(-0.716473\pi\)
−0.628847 + 0.777529i \(0.716473\pi\)
\(48\) 0 0
\(49\) −6.92750 −0.989642
\(50\) 0 0
\(51\) 0 0
\(52\) −2.62231 −0.363649
\(53\) 5.81940 0.799356 0.399678 − 0.916656i \(-0.369122\pi\)
0.399678 + 0.916656i \(0.369122\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.30519 0.174413
\(57\) 0 0
\(58\) −10.9819 −1.44199
\(59\) 12.0646 1.57067 0.785337 − 0.619069i \(-0.212490\pi\)
0.785337 + 0.619069i \(0.212490\pi\)
\(60\) 0 0
\(61\) 7.78315 0.996530 0.498265 − 0.867025i \(-0.333971\pi\)
0.498265 + 0.867025i \(0.333971\pi\)
\(62\) −4.23301 −0.537593
\(63\) 0 0
\(64\) −8.21489 −1.02686
\(65\) 0 0
\(66\) 0 0
\(67\) 0.269264 0.0328958 0.0164479 − 0.999865i \(-0.494764\pi\)
0.0164479 + 0.999865i \(0.494764\pi\)
\(68\) 21.0762 2.55586
\(69\) 0 0
\(70\) 0 0
\(71\) −3.30519 −0.392253 −0.196127 − 0.980579i \(-0.562836\pi\)
−0.196127 + 0.980579i \(0.562836\pi\)
\(72\) 0 0
\(73\) 3.69448 0.432407 0.216203 − 0.976348i \(-0.430633\pi\)
0.216203 + 0.976348i \(0.430633\pi\)
\(74\) −2.44579 −0.284317
\(75\) 0 0
\(76\) 12.2330 1.40322
\(77\) 1.60582 0.183000
\(78\) 0 0
\(79\) 8.85532 0.996302 0.498151 − 0.867090i \(-0.334013\pi\)
0.498151 + 0.867090i \(0.334013\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 17.5139 1.93408
\(83\) −5.34144 −0.586299 −0.293150 − 0.956067i \(-0.594703\pi\)
−0.293150 + 0.956067i \(0.594703\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −19.6945 −2.12371
\(87\) 0 0
\(88\) 28.9077 3.08157
\(89\) −5.20925 −0.552179 −0.276090 − 0.961132i \(-0.589039\pi\)
−0.276090 + 0.961132i \(0.589039\pi\)
\(90\) 0 0
\(91\) 0.177327 0.0185889
\(92\) −35.9315 −3.74612
\(93\) 0 0
\(94\) 21.0883 2.17510
\(95\) 0 0
\(96\) 0 0
\(97\) −12.0475 −1.22324 −0.611621 − 0.791151i \(-0.709482\pi\)
−0.611621 + 0.791151i \(0.709482\pi\)
\(98\) 16.9432 1.71152
\(99\) 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.bv.1.1 4
3.2 odd 2 2775.2.a.x.1.4 4
5.4 even 2 333.2.a.g.1.4 4
15.14 odd 2 111.2.a.b.1.1 ✓ 4
20.19 odd 2 5328.2.a.bs.1.3 4
60.59 even 2 1776.2.a.u.1.2 4
105.104 even 2 5439.2.a.u.1.1 4
120.29 odd 2 7104.2.a.cc.1.3 4
120.59 even 2 7104.2.a.cf.1.3 4
555.554 odd 2 4107.2.a.i.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
111.2.a.b.1.1 ✓ 4 15.14 odd 2
333.2.a.g.1.4 4 5.4 even 2
1776.2.a.u.1.2 4 60.59 even 2
2775.2.a.x.1.4 4 3.2 odd 2
4107.2.a.i.1.4 4 555.554 odd 2
5328.2.a.bs.1.3 4 20.19 odd 2
5439.2.a.u.1.1 4 105.104 even 2
7104.2.a.cc.1.3 4 120.29 odd 2
7104.2.a.cf.1.3 4 120.59 even 2
8325.2.a.bv.1.1 4 1.1 even 1 trivial