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 = [5,2,0,8,0,0,0,3,0,0,1,0,-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: \(5\)
Coefficient field: 5.5.528933.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 6x^{3} + 4x^{2} + 7x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2775)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.27987\) 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)
 
\(f(q)\) \(=\) \(q+1.15875 q^{2} -0.657292 q^{4} +4.20154 q^{7} -3.07914 q^{8} +2.64180 q^{11} +1.14034 q^{13} +4.86855 q^{14} -2.25338 q^{16} +6.14034 q^{17} +7.45785 q^{19} +3.06120 q^{22} +8.18313 q^{23} +1.32138 q^{26} -2.76164 q^{28} -0.834454 q^{29} -6.63209 q^{31} +3.54717 q^{32} +7.11514 q^{34} -1.00000 q^{37} +8.64180 q^{38} -2.15583 q^{41} -4.30589 q^{43} -1.73644 q^{44} +9.48223 q^{46} -0.113041 q^{47} +10.6530 q^{49} -0.749538 q^{52} +0.113041 q^{53} -12.9372 q^{56} -0.966926 q^{58} +1.01841 q^{59} -9.13952 q^{61} -7.68495 q^{62} +8.61707 q^{64} -6.71849 q^{67} -4.03600 q^{68} -8.17880 q^{71} -6.42185 q^{73} -1.15875 q^{74} -4.90198 q^{76} +11.0997 q^{77} +8.87970 q^{79} -2.49807 q^{82} +6.10107 q^{83} -4.98946 q^{86} -8.13450 q^{88} -15.6662 q^{89} +4.79120 q^{91} -5.37870 q^{92} -0.130987 q^{94} -4.51403 q^{97} +12.3441 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 8 q^{4} + 3 q^{8} + q^{11} - 14 q^{13} + 10 q^{14} + 10 q^{16} + 11 q^{17} + 10 q^{19} + 14 q^{22} + 4 q^{23} - 13 q^{26} - 3 q^{28} - 5 q^{29} - 3 q^{31} + 23 q^{32} - 3 q^{34} - 5 q^{37}+ \cdots + 35 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.15875 0.819362 0.409681 − 0.912229i \(-0.365640\pi\)
0.409681 + 0.912229i \(0.365640\pi\)
\(3\) 0 0
\(4\) −0.657292 −0.328646
\(5\) 0 0
\(6\) 0 0
\(7\) 4.20154 1.58803 0.794017 − 0.607896i \(-0.207986\pi\)
0.794017 + 0.607896i \(0.207986\pi\)
\(8\) −3.07914 −1.08864
\(9\) 0 0
\(10\) 0 0
\(11\) 2.64180 0.796534 0.398267 − 0.917270i \(-0.369612\pi\)
0.398267 + 0.917270i \(0.369612\pi\)
\(12\) 0 0
\(13\) 1.14034 0.316274 0.158137 − 0.987417i \(-0.449451\pi\)
0.158137 + 0.987417i \(0.449451\pi\)
\(14\) 4.86855 1.30117
\(15\) 0 0
\(16\) −2.25338 −0.563346
\(17\) 6.14034 1.48925 0.744626 − 0.667482i \(-0.232628\pi\)
0.744626 + 0.667482i \(0.232628\pi\)
\(18\) 0 0
\(19\) 7.45785 1.71095 0.855474 − 0.517846i \(-0.173266\pi\)
0.855474 + 0.517846i \(0.173266\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.06120 0.652650
\(23\) 8.18313 1.70630 0.853151 − 0.521665i \(-0.174689\pi\)
0.853151 + 0.521665i \(0.174689\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.32138 0.259143
\(27\) 0 0
\(28\) −2.76164 −0.521901
\(29\) −0.834454 −0.154954 −0.0774771 − 0.996994i \(-0.524686\pi\)
−0.0774771 + 0.996994i \(0.524686\pi\)
\(30\) 0 0
\(31\) −6.63209 −1.19116 −0.595579 − 0.803297i \(-0.703077\pi\)
−0.595579 + 0.803297i \(0.703077\pi\)
\(32\) 3.54717 0.627058
\(33\) 0 0
\(34\) 7.11514 1.22024
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 8.64180 1.40189
\(39\) 0 0
\(40\) 0 0
\(41\) −2.15583 −0.336684 −0.168342 − 0.985729i \(-0.553841\pi\)
−0.168342 + 0.985729i \(0.553841\pi\)
\(42\) 0 0
\(43\) −4.30589 −0.656642 −0.328321 − 0.944566i \(-0.606483\pi\)
−0.328321 + 0.944566i \(0.606483\pi\)
\(44\) −1.73644 −0.261778
\(45\) 0 0
\(46\) 9.48223 1.39808
\(47\) −0.113041 −0.0164887 −0.00824436 − 0.999966i \(-0.502624\pi\)
−0.00824436 + 0.999966i \(0.502624\pi\)
\(48\) 0 0
\(49\) 10.6530 1.52185
\(50\) 0 0
\(51\) 0 0
\(52\) −0.749538 −0.103942
\(53\) 0.113041 0.0155274 0.00776369 − 0.999970i \(-0.497529\pi\)
0.00776369 + 0.999970i \(0.497529\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −12.9372 −1.72880
\(57\) 0 0
\(58\) −0.966926 −0.126964
\(59\) 1.01841 0.132586 0.0662928 − 0.997800i \(-0.478883\pi\)
0.0662928 + 0.997800i \(0.478883\pi\)
\(60\) 0 0
\(61\) −9.13952 −1.17020 −0.585098 − 0.810963i \(-0.698944\pi\)
−0.585098 + 0.810963i \(0.698944\pi\)
\(62\) −7.68495 −0.975990
\(63\) 0 0
\(64\) 8.61707 1.07713
\(65\) 0 0
\(66\) 0 0
\(67\) −6.71849 −0.820794 −0.410397 − 0.911907i \(-0.634610\pi\)
−0.410397 + 0.911907i \(0.634610\pi\)
\(68\) −4.03600 −0.489436
\(69\) 0 0
\(70\) 0 0
\(71\) −8.17880 −0.970645 −0.485322 − 0.874335i \(-0.661298\pi\)
−0.485322 + 0.874335i \(0.661298\pi\)
\(72\) 0 0
\(73\) −6.42185 −0.751621 −0.375811 − 0.926697i \(-0.622636\pi\)
−0.375811 + 0.926697i \(0.622636\pi\)
\(74\) −1.15875 −0.134702
\(75\) 0 0
\(76\) −4.90198 −0.562296
\(77\) 11.0997 1.26492
\(78\) 0 0
\(79\) 8.87970 0.999045 0.499522 − 0.866301i \(-0.333509\pi\)
0.499522 + 0.866301i \(0.333509\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −2.49807 −0.275866
\(83\) 6.10107 0.669679 0.334839 − 0.942275i \(-0.391318\pi\)
0.334839 + 0.942275i \(0.391318\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.98946 −0.538027
\(87\) 0 0
\(88\) −8.13450 −0.867140
\(89\) −15.6662 −1.66061 −0.830306 − 0.557308i \(-0.811834\pi\)
−0.830306 + 0.557308i \(0.811834\pi\)
\(90\) 0 0
\(91\) 4.79120 0.502254
\(92\) −5.37870 −0.560769
\(93\) 0 0
\(94\) −0.130987 −0.0135102
\(95\) 0 0
\(96\) 0 0
\(97\) −4.51403 −0.458330 −0.229165 − 0.973388i \(-0.573600\pi\)
−0.229165 + 0.973388i \(0.573600\pi\)
\(98\) 12.3441 1.24695
\(99\) 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.cg.1.3 5
3.2 odd 2 2775.2.a.z.1.3 ✓ 5
5.4 even 2 8325.2.a.ca.1.3 5
15.14 odd 2 2775.2.a.bc.1.3 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2775.2.a.z.1.3 ✓ 5 3.2 odd 2
2775.2.a.bc.1.3 yes 5 15.14 odd 2
8325.2.a.ca.1.3 5 5.4 even 2
8325.2.a.cg.1.3 5 1.1 even 1 trivial