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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [648,2,Mod(325,648)] 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("648.325"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,0,0,0,8,4,0,4,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 325.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 648.325
Dual form 648.2.d.a.325.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.00000i) q^{2} +2.00000i q^{4} +2.00000i q^{5} +4.00000 q^{7} +(2.00000 - 2.00000i) q^{8} +(2.00000 - 2.00000i) q^{10} +3.00000i q^{11} +2.00000i q^{13} +(-4.00000 - 4.00000i) q^{14} -4.00000 q^{16} -5.00000 q^{17} +1.00000i q^{19} -4.00000 q^{20} +(3.00000 - 3.00000i) q^{22} -2.00000 q^{23} +1.00000 q^{25} +(2.00000 - 2.00000i) q^{26} +8.00000i q^{28} -4.00000 q^{31} +(4.00000 + 4.00000i) q^{32} +(5.00000 + 5.00000i) q^{34} +8.00000i q^{35} +2.00000i q^{37} +(1.00000 - 1.00000i) q^{38} +(4.00000 + 4.00000i) q^{40} +5.00000 q^{41} +11.0000i q^{43} -6.00000 q^{44} +(2.00000 + 2.00000i) q^{46} +6.00000 q^{47} +9.00000 q^{49} +(-1.00000 - 1.00000i) q^{50} -4.00000 q^{52} -6.00000 q^{55} +(8.00000 - 8.00000i) q^{56} +1.00000i q^{59} -12.0000i q^{61} +(4.00000 + 4.00000i) q^{62} -8.00000i q^{64} -4.00000 q^{65} +3.00000i q^{67} -10.0000i q^{68} +(8.00000 - 8.00000i) q^{70} +6.00000 q^{71} +9.00000 q^{73} +(2.00000 - 2.00000i) q^{74} -2.00000 q^{76} +12.0000i q^{77} -14.0000 q^{79} -8.00000i q^{80} +(-5.00000 - 5.00000i) q^{82} +4.00000i q^{83} -10.0000i q^{85} +(11.0000 - 11.0000i) q^{86} +(6.00000 + 6.00000i) q^{88} +14.0000 q^{89} +8.00000i q^{91} -4.00000i q^{92} +(-6.00000 - 6.00000i) q^{94} -2.00000 q^{95} +1.00000 q^{97} +(-9.00000 - 9.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 8 q^{7} + 4 q^{8} + 4 q^{10} - 8 q^{14} - 8 q^{16} - 10 q^{17} - 8 q^{20} + 6 q^{22} - 4 q^{23} + 2 q^{25} + 4 q^{26} - 8 q^{31} + 8 q^{32} + 10 q^{34} + 2 q^{38} + 8 q^{40} + 10 q^{41}+ \cdots - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 − 1.00000i −0.707107 − 0.707107i
\(3\) 0 0
\(4\) 2.00000i 1.00000i
\(5\) 2.00000i 0.894427i 0.894427 + 0.447214i \(0.147584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 2.00000 − 2.00000i 0.707107 − 0.707107i
\(9\) 0 0
\(10\) 2.00000 − 2.00000i 0.632456 − 0.632456i
\(11\) 3.00000i 0.904534i 0.891883 + 0.452267i \(0.149385\pi\)
−0.891883 + 0.452267i \(0.850615\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) −4.00000 − 4.00000i −1.06904 − 1.06904i
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −5.00000 −1.21268 −0.606339 − 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i 0.993399 + 0.114708i \(0.0365932\pi\)
−0.993399 + 0.114708i \(0.963407\pi\)
\(20\) −4.00000 −0.894427
\(21\) 0 0
\(22\) 3.00000 − 3.00000i 0.639602 − 0.639602i
\(23\) −2.00000 −0.417029 −0.208514 − 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 − 2.00000i 0.392232 − 0.392232i
\(27\) 0 0
\(28\) 8.00000i 1.51186i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 4.00000 + 4.00000i 0.707107 + 0.707107i
\(33\) 0 0
\(34\) 5.00000 + 5.00000i 0.857493 + 0.857493i
\(35\) 8.00000i 1.35225i
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 1.00000 − 1.00000i 0.162221 − 0.162221i
\(39\) 0 0
\(40\) 4.00000 + 4.00000i 0.632456 + 0.632456i
\(41\) 5.00000 0.780869 0.390434 − 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 11.0000i 1.67748i 0.544529 + 0.838742i \(0.316708\pi\)
−0.544529 + 0.838742i \(0.683292\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 2.00000 + 2.00000i 0.294884 + 0.294884i
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) −1.00000 − 1.00000i −0.141421 − 0.141421i
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) 8.00000 − 8.00000i 1.06904 − 1.06904i
\(57\) 0 0
\(58\) 0 0
\(59\) 1.00000i 0.130189i 0.997879 + 0.0650945i \(0.0207349\pi\)
−0.997879 + 0.0650945i \(0.979265\pi\)
\(60\) 0 0
\(61\) − 12.0000i − 1.53644i −0.640184 − 0.768221i \(-0.721142\pi\)
0.640184 − 0.768221i \(-0.278858\pi\)
\(62\) 4.00000 + 4.00000i 0.508001 + 0.508001i
\(63\) 0 0
\(64\) − 8.00000i − 1.00000i
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 3.00000i 0.366508i 0.983066 + 0.183254i \(0.0586631\pi\)
−0.983066 + 0.183254i \(0.941337\pi\)
\(68\) − 10.0000i − 1.21268i
\(69\) 0 0
\(70\) 8.00000 − 8.00000i 0.956183 − 0.956183i
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 − 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 2.00000 − 2.00000i 0.232495 − 0.232495i
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 12.0000i 1.36753i
\(78\) 0 0
\(79\) −14.0000 −1.57512 −0.787562 − 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) − 8.00000i − 0.894427i
\(81\) 0 0
\(82\) −5.00000 − 5.00000i −0.552158 − 0.552158i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) − 10.0000i − 1.08465i
\(86\) 11.0000 − 11.0000i 1.18616 − 1.18616i
\(87\) 0 0
\(88\) 6.00000 + 6.00000i 0.639602 + 0.639602i
\(89\) 14.0000 1.48400 0.741999 − 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 8.00000i 0.838628i
\(92\) − 4.00000i − 0.417029i
\(93\) 0 0
\(94\) −6.00000 − 6.00000i −0.618853 − 0.618853i
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) −9.00000 − 9.00000i −0.909137 − 0.909137i
\(99\) 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 648.2.d.a.325.1 2
3.2 odd 2 648.2.d.d.325.2 2
4.3 odd 2 2592.2.d.a.1297.2 2
8.3 odd 2 2592.2.d.a.1297.1 2
8.5 even 2 inner 648.2.d.a.325.2 2
9.2 odd 6 72.2.n.a.13.2 yes 4
9.4 even 3 216.2.n.a.181.2 4
9.5 odd 6 72.2.n.a.61.1 yes 4
9.7 even 3 216.2.n.a.37.1 4
12.11 even 2 2592.2.d.b.1297.1 2
24.5 odd 2 648.2.d.d.325.1 2
24.11 even 2 2592.2.d.b.1297.2 2
36.7 odd 6 864.2.r.a.145.1 4
36.11 even 6 288.2.r.a.49.1 4
36.23 even 6 288.2.r.a.241.2 4
36.31 odd 6 864.2.r.a.721.2 4
72.5 odd 6 72.2.n.a.61.2 yes 4
72.11 even 6 288.2.r.a.49.2 4
72.13 even 6 216.2.n.a.181.1 4
72.29 odd 6 72.2.n.a.13.1 ✓ 4
72.43 odd 6 864.2.r.a.145.2 4
72.59 even 6 288.2.r.a.241.1 4
72.61 even 6 216.2.n.a.37.2 4
72.67 odd 6 864.2.r.a.721.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.n.a.13.1 ✓ 4 72.29 odd 6
72.2.n.a.13.2 yes 4 9.2 odd 6
72.2.n.a.61.1 yes 4 9.5 odd 6
72.2.n.a.61.2 yes 4 72.5 odd 6
216.2.n.a.37.1 4 9.7 even 3
216.2.n.a.37.2 4 72.61 even 6
216.2.n.a.181.1 4 72.13 even 6
216.2.n.a.181.2 4 9.4 even 3
288.2.r.a.49.1 4 36.11 even 6
288.2.r.a.49.2 4 72.11 even 6
288.2.r.a.241.1 4 72.59 even 6
288.2.r.a.241.2 4 36.23 even 6
648.2.d.a.325.1 2 1.1 even 1 trivial
648.2.d.a.325.2 2 8.5 even 2 inner
648.2.d.d.325.1 2 24.5 odd 2
648.2.d.d.325.2 2 3.2 odd 2
864.2.r.a.145.1 4 36.7 odd 6
864.2.r.a.145.2 4 72.43 odd 6
864.2.r.a.721.1 4 72.67 odd 6
864.2.r.a.721.2 4 36.31 odd 6
2592.2.d.a.1297.1 2 8.3 odd 2
2592.2.d.a.1297.2 2 4.3 odd 2
2592.2.d.b.1297.1 2 12.11 even 2
2592.2.d.b.1297.2 2 24.11 even 2