Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [832,2,Mod(257,832)] 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("832.257"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(832, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 832 = 2^{6} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 832.w (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,0,0,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.64355344817\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 257.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 832.257
Dual form 832.2.w.d.641.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.73205i) q^{3} -1.73205i q^{5} +(-0.500000 + 0.866025i) q^{9} +(2.50000 - 2.59808i) q^{13} +(3.00000 - 1.73205i) q^{15} +(1.50000 - 2.59808i) q^{17} +(3.00000 + 1.73205i) q^{19} +(3.00000 + 5.19615i) q^{23} +2.00000 q^{25} +4.00000 q^{27} +(1.50000 + 2.59808i) q^{29} -3.46410i q^{31} +(-7.50000 + 4.33013i) q^{37} +(7.00000 + 1.73205i) q^{39} +(-4.50000 + 2.59808i) q^{41} +(4.00000 - 6.92820i) q^{43} +(1.50000 + 0.866025i) q^{45} +3.46410i q^{47} +(-3.50000 - 6.06218i) q^{49} +6.00000 q^{51} +3.00000 q^{53} +6.92820i q^{57} +(-6.00000 - 3.46410i) q^{59} +(0.500000 - 0.866025i) q^{61} +(-4.50000 - 4.33013i) q^{65} +(-3.00000 + 1.73205i) q^{67} +(-6.00000 + 10.3923i) q^{69} +(3.00000 + 1.73205i) q^{71} -1.73205i q^{73} +(2.00000 + 3.46410i) q^{75} +4.00000 q^{79} +(5.50000 + 9.52628i) q^{81} +13.8564i q^{83} +(-4.50000 - 2.59808i) q^{85} +(-3.00000 + 5.19615i) q^{87} +(-6.00000 + 3.46410i) q^{89} +(6.00000 - 3.46410i) q^{93} +(3.00000 - 5.19615i) q^{95} +(6.00000 + 3.46410i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{9} + 5 q^{13} + 6 q^{15} + 3 q^{17} + 6 q^{19} + 6 q^{23} + 4 q^{25} + 8 q^{27} + 3 q^{29} - 15 q^{37} + 14 q^{39} - 9 q^{41} + 8 q^{43} + 3 q^{45} - 7 q^{49} + 12 q^{51} + 6 q^{53}+ \cdots + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/832\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(703\) \(769\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 + 1.73205i 0.577350 + 1.00000i 0.995782 + 0.0917517i \(0.0292466\pi\)
−0.418432 + 0.908248i \(0.637420\pi\)
\(4\) 0 0
\(5\) − 1.73205i − 0.774597i −0.921954 − 0.387298i \(-0.873408\pi\)
0.921954 − 0.387298i \(-0.126592\pi\)
\(6\) 0 0
\(7\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) 2.50000 − 2.59808i 0.693375 − 0.720577i
\(14\) 0 0
\(15\) 3.00000 − 1.73205i 0.774597 − 0.447214i
\(16\) 0 0
\(17\) 1.50000 − 2.59808i 0.363803 − 0.630126i −0.624780 − 0.780801i \(-0.714811\pi\)
0.988583 + 0.150675i \(0.0481447\pi\)
\(18\) 0 0
\(19\) 3.00000 + 1.73205i 0.688247 + 0.397360i 0.802955 − 0.596040i \(-0.203260\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 + 5.19615i 0.625543 + 1.08347i 0.988436 + 0.151642i \(0.0484560\pi\)
−0.362892 + 0.931831i \(0.618211\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 1.50000 + 2.59808i 0.278543 + 0.482451i 0.971023 − 0.238987i \(-0.0768152\pi\)
−0.692480 + 0.721437i \(0.743482\pi\)
\(30\) 0 0
\(31\) − 3.46410i − 0.622171i −0.950382 − 0.311086i \(-0.899307\pi\)
0.950382 − 0.311086i \(-0.100693\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.50000 + 4.33013i −1.23299 + 0.711868i −0.967653 − 0.252286i \(-0.918817\pi\)
−0.265340 + 0.964155i \(0.585484\pi\)
\(38\) 0 0
\(39\) 7.00000 + 1.73205i 1.12090 + 0.277350i
\(40\) 0 0
\(41\) −4.50000 + 2.59808i −0.702782 + 0.405751i −0.808383 − 0.588657i \(-0.799657\pi\)
0.105601 + 0.994409i \(0.466323\pi\)
\(42\) 0 0
\(43\) 4.00000 − 6.92820i 0.609994 − 1.05654i −0.381246 − 0.924473i \(-0.624505\pi\)
0.991241 − 0.132068i \(-0.0421616\pi\)
\(44\) 0 0
\(45\) 1.50000 + 0.866025i 0.223607 + 0.129099i
\(46\) 0 0
\(47\) 3.46410i 0.505291i 0.967559 + 0.252646i \(0.0813007\pi\)
−0.967559 + 0.252646i \(0.918699\pi\)
\(48\) 0 0
\(49\) −3.50000 − 6.06218i −0.500000 − 0.866025i
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 − 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.92820i 0.917663i
\(58\) 0 0
\(59\) −6.00000 − 3.46410i −0.781133 − 0.450988i 0.0556984 − 0.998448i \(-0.482261\pi\)
−0.836832 + 0.547460i \(0.815595\pi\)
\(60\) 0 0
\(61\) 0.500000 − 0.866025i 0.0640184 − 0.110883i −0.832240 − 0.554416i \(-0.812942\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.50000 − 4.33013i −0.558156 − 0.537086i
\(66\) 0 0
\(67\) −3.00000 + 1.73205i −0.366508 + 0.211604i −0.671932 − 0.740613i \(-0.734535\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 0 0
\(69\) −6.00000 + 10.3923i −0.722315 + 1.25109i
\(70\) 0 0
\(71\) 3.00000 + 1.73205i 0.356034 + 0.205557i 0.667340 − 0.744753i \(-0.267433\pi\)
−0.311305 + 0.950310i \(0.600766\pi\)
\(72\) 0 0
\(73\) − 1.73205i − 0.202721i −0.994850 − 0.101361i \(-0.967680\pi\)
0.994850 − 0.101361i \(-0.0323196\pi\)
\(74\) 0 0
\(75\) 2.00000 + 3.46410i 0.230940 + 0.400000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 5.50000 + 9.52628i 0.611111 + 1.05848i
\(82\) 0 0
\(83\) 13.8564i 1.52094i 0.649374 + 0.760469i \(0.275031\pi\)
−0.649374 + 0.760469i \(0.724969\pi\)
\(84\) 0 0
\(85\) −4.50000 − 2.59808i −0.488094 − 0.281801i
\(86\) 0 0
\(87\) −3.00000 + 5.19615i −0.321634 + 0.557086i
\(88\) 0 0
\(89\) −6.00000 + 3.46410i −0.635999 + 0.367194i −0.783072 − 0.621932i \(-0.786348\pi\)
0.147073 + 0.989126i \(0.453015\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.00000 − 3.46410i 0.622171 − 0.359211i
\(94\) 0 0
\(95\) 3.00000 − 5.19615i 0.307794 − 0.533114i
\(96\) 0 0
\(97\) 6.00000 + 3.46410i 0.609208 + 0.351726i 0.772655 − 0.634826i \(-0.218928\pi\)
−0.163448 + 0.986552i \(0.552261\pi\)
\(98\) 0 0
\(99\) 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 832.2.w.d.257.1 2
4.3 odd 2 832.2.w.a.257.1 2
8.3 odd 2 208.2.w.b.49.1 2
8.5 even 2 13.2.e.a.10.1 yes 2
13.4 even 6 inner 832.2.w.d.641.1 2
24.5 odd 2 117.2.q.c.10.1 2
24.11 even 2 1872.2.by.d.1297.1 2
40.13 odd 4 325.2.m.a.49.2 4
40.29 even 2 325.2.n.a.101.1 2
40.37 odd 4 325.2.m.a.49.1 4
52.43 odd 6 832.2.w.a.641.1 2
56.5 odd 6 637.2.u.b.361.1 2
56.13 odd 2 637.2.q.a.491.1 2
56.37 even 6 637.2.u.c.361.1 2
56.45 odd 6 637.2.k.c.569.1 2
56.53 even 6 637.2.k.a.569.1 2
104.3 odd 6 2704.2.f.b.337.2 2
104.5 odd 4 169.2.c.a.146.2 4
104.11 even 12 2704.2.a.o.1.1 2
104.21 odd 4 169.2.c.a.146.1 4
104.29 even 6 169.2.b.a.168.1 2
104.37 odd 12 169.2.a.a.1.2 2
104.43 odd 6 208.2.w.b.17.1 2
104.45 odd 12 169.2.c.a.22.2 4
104.61 even 6 169.2.e.a.147.1 2
104.67 even 12 2704.2.a.o.1.2 2
104.69 even 6 13.2.e.a.4.1 ✓ 2
104.75 odd 6 2704.2.f.b.337.1 2
104.77 even 2 169.2.e.a.23.1 2
104.85 odd 12 169.2.c.a.22.1 4
104.93 odd 12 169.2.a.a.1.1 2
104.101 even 6 169.2.b.a.168.2 2
312.29 odd 6 1521.2.b.a.1351.2 2
312.101 odd 6 1521.2.b.a.1351.1 2
312.173 odd 6 117.2.q.c.82.1 2
312.197 even 12 1521.2.a.k.1.2 2
312.245 even 12 1521.2.a.k.1.1 2
312.251 even 6 1872.2.by.d.433.1 2
520.69 even 6 325.2.n.a.251.1 2
520.173 odd 12 325.2.m.a.199.1 4
520.277 odd 12 325.2.m.a.199.2 4
520.349 odd 12 4225.2.a.v.1.1 2
520.509 odd 12 4225.2.a.v.1.2 2
728.69 odd 6 637.2.q.a.589.1 2
728.173 odd 6 637.2.k.c.459.1 2
728.277 even 6 637.2.u.c.30.1 2
728.349 even 12 8281.2.a.q.1.2 2
728.381 odd 6 637.2.u.b.30.1 2
728.405 even 12 8281.2.a.q.1.1 2
728.485 even 6 637.2.k.a.459.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.2.e.a.4.1 ✓ 2 104.69 even 6
13.2.e.a.10.1 yes 2 8.5 even 2
117.2.q.c.10.1 2 24.5 odd 2
117.2.q.c.82.1 2 312.173 odd 6
169.2.a.a.1.1 2 104.93 odd 12
169.2.a.a.1.2 2 104.37 odd 12
169.2.b.a.168.1 2 104.29 even 6
169.2.b.a.168.2 2 104.101 even 6
169.2.c.a.22.1 4 104.85 odd 12
169.2.c.a.22.2 4 104.45 odd 12
169.2.c.a.146.1 4 104.21 odd 4
169.2.c.a.146.2 4 104.5 odd 4
169.2.e.a.23.1 2 104.77 even 2
169.2.e.a.147.1 2 104.61 even 6
208.2.w.b.17.1 2 104.43 odd 6
208.2.w.b.49.1 2 8.3 odd 2
325.2.m.a.49.1 4 40.37 odd 4
325.2.m.a.49.2 4 40.13 odd 4
325.2.m.a.199.1 4 520.173 odd 12
325.2.m.a.199.2 4 520.277 odd 12
325.2.n.a.101.1 2 40.29 even 2
325.2.n.a.251.1 2 520.69 even 6
637.2.k.a.459.1 2 728.485 even 6
637.2.k.a.569.1 2 56.53 even 6
637.2.k.c.459.1 2 728.173 odd 6
637.2.k.c.569.1 2 56.45 odd 6
637.2.q.a.491.1 2 56.13 odd 2
637.2.q.a.589.1 2 728.69 odd 6
637.2.u.b.30.1 2 728.381 odd 6
637.2.u.b.361.1 2 56.5 odd 6
637.2.u.c.30.1 2 728.277 even 6
637.2.u.c.361.1 2 56.37 even 6
832.2.w.a.257.1 2 4.3 odd 2
832.2.w.a.641.1 2 52.43 odd 6
832.2.w.d.257.1 2 1.1 even 1 trivial
832.2.w.d.641.1 2 13.4 even 6 inner
1521.2.a.k.1.1 2 312.245 even 12
1521.2.a.k.1.2 2 312.197 even 12
1521.2.b.a.1351.1 2 312.101 odd 6
1521.2.b.a.1351.2 2 312.29 odd 6
1872.2.by.d.433.1 2 312.251 even 6
1872.2.by.d.1297.1 2 24.11 even 2
2704.2.a.o.1.1 2 104.11 even 12
2704.2.a.o.1.2 2 104.67 even 12
2704.2.f.b.337.1 2 104.75 odd 6
2704.2.f.b.337.2 2 104.3 odd 6
4225.2.a.v.1.1 2 520.349 odd 12
4225.2.a.v.1.2 2 520.509 odd 12
8281.2.a.q.1.1 2 728.405 even 12
8281.2.a.q.1.2 2 728.349 even 12