Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [48,18,Mod(1,48)] 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("48.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(48, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-13122,0,382860] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(87.9466019254\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14569}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3642 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-59.8511\) of defining polynomial
Character \(\chi\) \(=\) 48.1

$q$-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)
 
\(f(q)\) \(=\) \(q-6561.00 q^{3} +1.46604e6 q^{5} -2.28730e7 q^{7} +4.30467e7 q^{9} +5.40173e8 q^{11} -3.45398e7 q^{13} -9.61872e9 q^{15} -9.43866e9 q^{17} +2.25061e9 q^{19} +1.50070e11 q^{21} +3.45854e11 q^{23} +1.38635e12 q^{25} -2.82430e11 q^{27} +5.11838e11 q^{29} -1.14436e11 q^{31} -3.54407e12 q^{33} -3.35329e13 q^{35} -1.56972e13 q^{37} +2.26616e11 q^{39} -8.00761e13 q^{41} +3.66737e13 q^{43} +6.31084e13 q^{45} +1.17577e14 q^{47} +2.90545e14 q^{49} +6.19270e13 q^{51} -3.90043e14 q^{53} +7.91917e14 q^{55} -1.47663e13 q^{57} -1.82562e15 q^{59} +1.41053e15 q^{61} -9.84609e14 q^{63} -5.06369e13 q^{65} +1.47114e15 q^{67} -2.26915e15 q^{69} +7.31441e15 q^{71} +1.34580e16 q^{73} -9.09582e15 q^{75} -1.23554e16 q^{77} +8.37779e15 q^{79} +1.85302e15 q^{81} +2.55978e16 q^{83} -1.38375e16 q^{85} -3.35817e15 q^{87} +4.47540e16 q^{89} +7.90031e14 q^{91} +7.50815e14 q^{93} +3.29949e15 q^{95} +7.41658e16 q^{97} +2.32527e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 13122 q^{3} + 382860 q^{5} - 24471568 q^{7} + 86093442 q^{9} + 987553512 q^{11} - 2519398244 q^{13} - 2511944460 q^{15} - 34313126364 q^{17} - 80053542184 q^{19} + 160557957648 q^{21} - 297228742704 q^{23}+ \cdots + 42\!\cdots\!52 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −6561.00 −0.577350
\(4\) 0 0
\(5\) 1.46604e6 1.67843 0.839213 − 0.543803i \(-0.183016\pi\)
0.839213 + 0.543803i \(0.183016\pi\)
\(6\) 0 0
\(7\) −2.28730e7 −1.49965 −0.749825 − 0.661636i \(-0.769863\pi\)
−0.749825 + 0.661636i \(0.769863\pi\)
\(8\) 0 0
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) 5.40173e8 0.759792 0.379896 − 0.925029i \(-0.375960\pi\)
0.379896 + 0.925029i \(0.375960\pi\)
\(12\) 0 0
\(13\) −3.45398e7 −0.0117436 −0.00587181 − 0.999983i \(-0.501869\pi\)
−0.00587181 + 0.999983i \(0.501869\pi\)
\(14\) 0 0
\(15\) −9.61872e9 −0.969039
\(16\) 0 0
\(17\) −9.43866e9 −0.328167 −0.164083 − 0.986446i \(-0.552467\pi\)
−0.164083 + 0.986446i \(0.552467\pi\)
\(18\) 0 0
\(19\) 2.25061e9 0.0304015 0.0152007 − 0.999884i \(-0.495161\pi\)
0.0152007 + 0.999884i \(0.495161\pi\)
\(20\) 0 0
\(21\) 1.50070e11 0.865824
\(22\) 0 0
\(23\) 3.45854e11 0.920886 0.460443 − 0.887689i \(-0.347690\pi\)
0.460443 + 0.887689i \(0.347690\pi\)
\(24\) 0 0
\(25\) 1.38635e12 1.81711
\(26\) 0 0
\(27\) −2.82430e11 −0.192450
\(28\) 0 0
\(29\) 5.11838e11 0.189998 0.0949992 − 0.995477i \(-0.469715\pi\)
0.0949992 + 0.995477i \(0.469715\pi\)
\(30\) 0 0
\(31\) −1.14436e11 −0.0240984 −0.0120492 − 0.999927i \(-0.503835\pi\)
−0.0120492 + 0.999927i \(0.503835\pi\)
\(32\) 0 0
\(33\) −3.54407e12 −0.438666
\(34\) 0 0
\(35\) −3.35329e13 −2.51705
\(36\) 0 0
\(37\) −1.56972e13 −0.734697 −0.367349 − 0.930083i \(-0.619734\pi\)
−0.367349 + 0.930083i \(0.619734\pi\)
\(38\) 0 0
\(39\) 2.26616e11 0.00678018
\(40\) 0 0
\(41\) −8.00761e13 −1.56617 −0.783087 − 0.621912i \(-0.786356\pi\)
−0.783087 + 0.621912i \(0.786356\pi\)
\(42\) 0 0
\(43\) 3.66737e13 0.478489 0.239245 − 0.970959i \(-0.423100\pi\)
0.239245 + 0.970959i \(0.423100\pi\)
\(44\) 0 0
\(45\) 6.31084e13 0.559475
\(46\) 0 0
\(47\) 1.17577e14 0.720263 0.360132 − 0.932901i \(-0.382732\pi\)
0.360132 + 0.932901i \(0.382732\pi\)
\(48\) 0 0
\(49\) 2.90545e14 1.24895
\(50\) 0 0
\(51\) 6.19270e13 0.189467
\(52\) 0 0
\(53\) −3.90043e14 −0.860534 −0.430267 − 0.902702i \(-0.641581\pi\)
−0.430267 + 0.902702i \(0.641581\pi\)
\(54\) 0 0
\(55\) 7.91917e14 1.27525
\(56\) 0 0
\(57\) −1.47663e13 −0.0175523
\(58\) 0 0
\(59\) −1.82562e15 −1.61870 −0.809352 − 0.587323i \(-0.800182\pi\)
−0.809352 + 0.587323i \(0.800182\pi\)
\(60\) 0 0
\(61\) 1.41053e15 0.942059 0.471029 − 0.882118i \(-0.343883\pi\)
0.471029 + 0.882118i \(0.343883\pi\)
\(62\) 0 0
\(63\) −9.84609e14 −0.499884
\(64\) 0 0
\(65\) −5.06369e13 −0.0197108
\(66\) 0 0
\(67\) 1.47114e15 0.442607 0.221304 − 0.975205i \(-0.428969\pi\)
0.221304 + 0.975205i \(0.428969\pi\)
\(68\) 0 0
\(69\) −2.26915e15 −0.531674
\(70\) 0 0
\(71\) 7.31441e15 1.34426 0.672130 − 0.740433i \(-0.265380\pi\)
0.672130 + 0.740433i \(0.265380\pi\)
\(72\) 0 0
\(73\) 1.34580e16 1.95315 0.976577 − 0.215169i \(-0.0690302\pi\)
0.976577 + 0.215169i \(0.0690302\pi\)
\(74\) 0 0
\(75\) −9.09582e15 −1.04911
\(76\) 0 0
\(77\) −1.23554e16 −1.13942
\(78\) 0 0
\(79\) 8.37779e15 0.621297 0.310648 − 0.950525i \(-0.399454\pi\)
0.310648 + 0.950525i \(0.399454\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 0 0
\(83\) 2.55978e16 1.24750 0.623748 − 0.781625i \(-0.285609\pi\)
0.623748 + 0.781625i \(0.285609\pi\)
\(84\) 0 0
\(85\) −1.38375e16 −0.550803
\(86\) 0 0
\(87\) −3.35817e15 −0.109696
\(88\) 0 0
\(89\) 4.47540e16 1.20508 0.602541 − 0.798088i \(-0.294155\pi\)
0.602541 + 0.798088i \(0.294155\pi\)
\(90\) 0 0
\(91\) 7.90031e14 0.0176113
\(92\) 0 0
\(93\) 7.50815e14 0.0139132
\(94\) 0 0
\(95\) 3.29949e15 0.0510266
\(96\) 0 0
\(97\) 7.41658e16 0.960824 0.480412 − 0.877043i \(-0.340487\pi\)
0.480412 + 0.877043i \(0.340487\pi\)
\(98\) 0 0
\(99\) 2.32527e16 0.253264
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 48.18.a.h.1.2 2
4.3 odd 2 3.18.a.b.1.1 ✓ 2
12.11 even 2 9.18.a.c.1.2 2
20.3 even 4 75.18.b.c.49.3 4
20.7 even 4 75.18.b.c.49.2 4
20.19 odd 2 75.18.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.18.a.b.1.1 ✓ 2 4.3 odd 2
9.18.a.c.1.2 2 12.11 even 2
48.18.a.h.1.2 2 1.1 even 1 trivial
75.18.a.b.1.2 2 20.19 odd 2
75.18.b.c.49.2 4 20.7 even 4
75.18.b.c.49.3 4 20.3 even 4