Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [325,2,Mod(49,325)] 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("325.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.m (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 199.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 325.199
Dual form 325.2.m.a.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 - 1.50000i) q^{2} +(1.73205 + 1.00000i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(3.00000 - 1.73205i) q^{6} +1.73205 q^{8} +(0.500000 + 0.866025i) q^{9} -2.00000i q^{12} +(-2.59808 + 2.50000i) q^{13} +(2.50000 - 4.33013i) q^{16} +(-2.59808 + 1.50000i) q^{17} +1.73205 q^{18} +(3.00000 - 1.73205i) q^{19} +(-5.19615 - 3.00000i) q^{23} +(3.00000 + 1.73205i) q^{24} +(1.50000 + 6.06218i) q^{26} -4.00000i q^{27} +(1.50000 - 2.59808i) q^{29} +3.46410i q^{31} +(-2.59808 - 4.50000i) q^{32} +5.19615i q^{34} +(0.500000 - 0.866025i) q^{36} +(-4.33013 + 7.50000i) q^{37} -6.00000i q^{38} +(-7.00000 + 1.73205i) q^{39} +(-4.50000 - 2.59808i) q^{41} +(-6.92820 + 4.00000i) q^{43} +(-9.00000 + 5.19615i) q^{46} +3.46410 q^{47} +(8.66025 - 5.00000i) q^{48} +(3.50000 - 6.06218i) q^{49} -6.00000 q^{51} +(3.46410 + 1.00000i) q^{52} +3.00000i q^{53} +(-6.00000 - 3.46410i) q^{54} +6.92820 q^{57} +(-2.59808 - 4.50000i) q^{58} +(-6.00000 + 3.46410i) q^{59} +(-0.500000 - 0.866025i) q^{61} +(5.19615 + 3.00000i) q^{62} +1.00000 q^{64} +(-1.73205 + 3.00000i) q^{67} +(2.59808 + 1.50000i) q^{68} +(-6.00000 - 10.3923i) q^{69} +(3.00000 - 1.73205i) q^{71} +(0.866025 + 1.50000i) q^{72} +1.73205 q^{73} +(7.50000 + 12.9904i) q^{74} +(-3.00000 - 1.73205i) q^{76} +(-3.46410 + 12.0000i) q^{78} -4.00000 q^{79} +(5.50000 - 9.52628i) q^{81} +(-7.79423 + 4.50000i) q^{82} +13.8564 q^{83} +13.8564i q^{86} +(5.19615 - 3.00000i) q^{87} +(6.00000 + 3.46410i) q^{89} +6.00000i q^{92} +(-3.46410 + 6.00000i) q^{93} +(3.00000 - 5.19615i) q^{94} -10.3923i q^{96} +(3.46410 + 6.00000i) q^{97} +(-6.06218 - 10.5000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + 12 q^{6} + 2 q^{9} + 10 q^{16} + 12 q^{19} + 12 q^{24} + 6 q^{26} + 6 q^{29} + 2 q^{36} - 28 q^{39} - 18 q^{41} - 36 q^{46} + 14 q^{49} - 24 q^{51} - 24 q^{54} - 24 q^{59} - 2 q^{61} + 4 q^{64}+ \cdots + 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 − 1.50000i 0.612372 − 1.06066i −0.378467 − 0.925615i \(-0.623549\pi\)
0.990839 − 0.135045i \(-0.0431180\pi\)
\(3\) 1.73205 + 1.00000i 1.00000 + 0.577350i 0.908248 − 0.418432i \(-0.137420\pi\)
0.0917517 + 0.995782i \(0.470753\pi\)
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 0 0
\(6\) 3.00000 − 1.73205i 1.22474 − 0.707107i
\(7\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 1.73205 0.612372
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) − 2.00000i − 0.577350i
\(13\) −2.59808 + 2.50000i −0.720577 + 0.693375i
\(14\) 0 0
\(15\) 0 0
\(16\) 2.50000 − 4.33013i 0.625000 − 1.08253i
\(17\) −2.59808 + 1.50000i −0.630126 + 0.363803i −0.780801 − 0.624780i \(-0.785189\pi\)
0.150675 + 0.988583i \(0.451855\pi\)
\(18\) 1.73205 0.408248
\(19\) 3.00000 − 1.73205i 0.688247 − 0.397360i −0.114708 − 0.993399i \(-0.536593\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.19615 − 3.00000i −1.08347 − 0.625543i −0.151642 − 0.988436i \(-0.548456\pi\)
−0.931831 + 0.362892i \(0.881789\pi\)
\(24\) 3.00000 + 1.73205i 0.612372 + 0.353553i
\(25\) 0 0
\(26\) 1.50000 + 6.06218i 0.294174 + 1.18889i
\(27\) − 4.00000i − 0.769800i
\(28\) 0 0
\(29\) 1.50000 − 2.59808i 0.278543 − 0.482451i −0.692480 − 0.721437i \(-0.743482\pi\)
0.971023 + 0.238987i \(0.0768152\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i 0.950382 + 0.311086i \(0.100693\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) −2.59808 − 4.50000i −0.459279 − 0.795495i
\(33\) 0 0
\(34\) 5.19615i 0.891133i
\(35\) 0 0
\(36\) 0.500000 − 0.866025i 0.0833333 − 0.144338i
\(37\) −4.33013 + 7.50000i −0.711868 + 1.23299i 0.252286 + 0.967653i \(0.418817\pi\)
−0.964155 + 0.265340i \(0.914516\pi\)
\(38\) − 6.00000i − 0.973329i
\(39\) −7.00000 + 1.73205i −1.12090 + 0.277350i
\(40\) 0 0
\(41\) −4.50000 − 2.59808i −0.702782 − 0.405751i 0.105601 − 0.994409i \(-0.466323\pi\)
−0.808383 + 0.588657i \(0.799657\pi\)
\(42\) 0 0
\(43\) −6.92820 + 4.00000i −1.05654 + 0.609994i −0.924473 − 0.381246i \(-0.875495\pi\)
−0.132068 + 0.991241i \(0.542162\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −9.00000 + 5.19615i −1.32698 + 0.766131i
\(47\) 3.46410 0.505291 0.252646 − 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) 8.66025 − 5.00000i 1.25000 − 0.721688i
\(49\) 3.50000 − 6.06218i 0.500000 − 0.866025i
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 3.46410 + 1.00000i 0.480384 + 0.138675i
\(53\) 3.00000i 0.412082i 0.978543 + 0.206041i \(0.0660580\pi\)
−0.978543 + 0.206041i \(0.933942\pi\)
\(54\) −6.00000 − 3.46410i −0.816497 − 0.471405i
\(55\) 0 0
\(56\) 0 0
\(57\) 6.92820 0.917663
\(58\) −2.59808 − 4.50000i −0.341144 − 0.590879i
\(59\) −6.00000 + 3.46410i −0.781133 + 0.450988i −0.836832 − 0.547460i \(-0.815595\pi\)
0.0556984 + 0.998448i \(0.482261\pi\)
\(60\) 0 0
\(61\) −0.500000 − 0.866025i −0.0640184 − 0.110883i 0.832240 − 0.554416i \(-0.187058\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 5.19615 + 3.00000i 0.659912 + 0.381000i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.73205 + 3.00000i −0.211604 + 0.366508i −0.952217 − 0.305424i \(-0.901202\pi\)
0.740613 + 0.671932i \(0.234535\pi\)
\(68\) 2.59808 + 1.50000i 0.315063 + 0.181902i
\(69\) −6.00000 − 10.3923i −0.722315 − 1.25109i
\(70\) 0 0
\(71\) 3.00000 − 1.73205i 0.356034 − 0.205557i −0.311305 − 0.950310i \(-0.600766\pi\)
0.667340 + 0.744753i \(0.267433\pi\)
\(72\) 0.866025 + 1.50000i 0.102062 + 0.176777i
\(73\) 1.73205 0.202721 0.101361 − 0.994850i \(-0.467680\pi\)
0.101361 + 0.994850i \(0.467680\pi\)
\(74\) 7.50000 + 12.9904i 0.871857 + 1.51010i
\(75\) 0 0
\(76\) −3.00000 − 1.73205i −0.344124 − 0.198680i
\(77\) 0 0
\(78\) −3.46410 + 12.0000i −0.392232 + 1.35873i
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 5.50000 − 9.52628i 0.611111 − 1.05848i
\(82\) −7.79423 + 4.50000i −0.860729 + 0.496942i
\(83\) 13.8564 1.52094 0.760469 − 0.649374i \(-0.224969\pi\)
0.760469 + 0.649374i \(0.224969\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 13.8564i 1.49417i
\(87\) 5.19615 − 3.00000i 0.557086 − 0.321634i
\(88\) 0 0
\(89\) 6.00000 + 3.46410i 0.635999 + 0.367194i 0.783072 − 0.621932i \(-0.213652\pi\)
−0.147073 + 0.989126i \(0.546985\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.00000i 0.625543i
\(93\) −3.46410 + 6.00000i −0.359211 + 0.622171i
\(94\) 3.00000 − 5.19615i 0.309426 − 0.535942i
\(95\) 0 0
\(96\) − 10.3923i − 1.06066i
\(97\) 3.46410 + 6.00000i 0.351726 + 0.609208i 0.986552 − 0.163448i \(-0.0522615\pi\)
−0.634826 + 0.772655i \(0.718928\pi\)
\(98\) −6.06218 − 10.5000i −0.612372 − 1.06066i
\(99\) 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 325.2.m.a.199.2 4
5.2 odd 4 325.2.n.a.251.1 2
5.3 odd 4 13.2.e.a.4.1 ✓ 2
5.4 even 2 inner 325.2.m.a.199.1 4
13.10 even 6 inner 325.2.m.a.49.1 4
15.8 even 4 117.2.q.c.82.1 2
20.3 even 4 208.2.w.b.17.1 2
35.3 even 12 637.2.u.b.30.1 2
35.13 even 4 637.2.q.a.589.1 2
35.18 odd 12 637.2.u.c.30.1 2
35.23 odd 12 637.2.k.a.459.1 2
35.33 even 12 637.2.k.c.459.1 2
40.3 even 4 832.2.w.a.641.1 2
40.13 odd 4 832.2.w.d.641.1 2
60.23 odd 4 1872.2.by.d.433.1 2
65.3 odd 12 169.2.e.a.23.1 2
65.7 even 12 4225.2.a.v.1.2 2
65.8 even 4 169.2.c.a.22.2 4
65.18 even 4 169.2.c.a.22.1 4
65.23 odd 12 13.2.e.a.10.1 yes 2
65.28 even 12 169.2.c.a.146.1 4
65.32 even 12 4225.2.a.v.1.1 2
65.33 even 12 169.2.a.a.1.1 2
65.38 odd 4 169.2.e.a.147.1 2
65.43 odd 12 169.2.b.a.168.1 2
65.48 odd 12 169.2.b.a.168.2 2
65.49 even 6 inner 325.2.m.a.49.2 4
65.58 even 12 169.2.a.a.1.2 2
65.62 odd 12 325.2.n.a.101.1 2
65.63 even 12 169.2.c.a.146.2 4
195.23 even 12 117.2.q.c.10.1 2
195.98 odd 12 1521.2.a.k.1.2 2
195.113 even 12 1521.2.b.a.1351.1 2
195.173 even 12 1521.2.b.a.1351.2 2
195.188 odd 12 1521.2.a.k.1.1 2
260.23 even 12 208.2.w.b.49.1 2
260.43 even 12 2704.2.f.b.337.2 2
260.123 odd 12 2704.2.a.o.1.1 2
260.163 odd 12 2704.2.a.o.1.2 2
260.243 even 12 2704.2.f.b.337.1 2
455.23 odd 12 637.2.u.c.361.1 2
455.88 odd 12 637.2.k.a.569.1 2
455.153 even 12 637.2.q.a.491.1 2
455.188 odd 12 8281.2.a.q.1.2 2
455.283 even 12 637.2.k.c.569.1 2
455.293 odd 12 8281.2.a.q.1.1 2
455.348 even 12 637.2.u.b.361.1 2
520.283 even 12 832.2.w.a.257.1 2
520.413 odd 12 832.2.w.d.257.1 2
780.23 odd 12 1872.2.by.d.1297.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.2.e.a.4.1 ✓ 2 5.3 odd 4
13.2.e.a.10.1 yes 2 65.23 odd 12
117.2.q.c.10.1 2 195.23 even 12
117.2.q.c.82.1 2 15.8 even 4
169.2.a.a.1.1 2 65.33 even 12
169.2.a.a.1.2 2 65.58 even 12
169.2.b.a.168.1 2 65.43 odd 12
169.2.b.a.168.2 2 65.48 odd 12
169.2.c.a.22.1 4 65.18 even 4
169.2.c.a.22.2 4 65.8 even 4
169.2.c.a.146.1 4 65.28 even 12
169.2.c.a.146.2 4 65.63 even 12
169.2.e.a.23.1 2 65.3 odd 12
169.2.e.a.147.1 2 65.38 odd 4
208.2.w.b.17.1 2 20.3 even 4
208.2.w.b.49.1 2 260.23 even 12
325.2.m.a.49.1 4 13.10 even 6 inner
325.2.m.a.49.2 4 65.49 even 6 inner
325.2.m.a.199.1 4 5.4 even 2 inner
325.2.m.a.199.2 4 1.1 even 1 trivial
325.2.n.a.101.1 2 65.62 odd 12
325.2.n.a.251.1 2 5.2 odd 4
637.2.k.a.459.1 2 35.23 odd 12
637.2.k.a.569.1 2 455.88 odd 12
637.2.k.c.459.1 2 35.33 even 12
637.2.k.c.569.1 2 455.283 even 12
637.2.q.a.491.1 2 455.153 even 12
637.2.q.a.589.1 2 35.13 even 4
637.2.u.b.30.1 2 35.3 even 12
637.2.u.b.361.1 2 455.348 even 12
637.2.u.c.30.1 2 35.18 odd 12
637.2.u.c.361.1 2 455.23 odd 12
832.2.w.a.257.1 2 520.283 even 12
832.2.w.a.641.1 2 40.3 even 4
832.2.w.d.257.1 2 520.413 odd 12
832.2.w.d.641.1 2 40.13 odd 4
1521.2.a.k.1.1 2 195.188 odd 12
1521.2.a.k.1.2 2 195.98 odd 12
1521.2.b.a.1351.1 2 195.113 even 12
1521.2.b.a.1351.2 2 195.173 even 12
1872.2.by.d.433.1 2 60.23 odd 4
1872.2.by.d.1297.1 2 780.23 odd 12
2704.2.a.o.1.1 2 260.123 odd 12
2704.2.a.o.1.2 2 260.163 odd 12
2704.2.f.b.337.1 2 260.243 even 12
2704.2.f.b.337.2 2 260.43 even 12
4225.2.a.v.1.1 2 65.32 even 12
4225.2.a.v.1.2 2 65.7 even 12
8281.2.a.q.1.1 2 455.293 odd 12
8281.2.a.q.1.2 2 455.188 odd 12