Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,18,Mod(1,9)] 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("9.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-594] 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: yes
Analytic conductor: \(16.4899878610\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
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 \(\chi\) \(=\) 9.1

$q$-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)
 
\(f(q)\) \(=\) \(q+65.1063 q^{2} -126833. q^{4} -1.46604e6 q^{5} +2.28730e7 q^{7} -1.67913e7 q^{8} -9.54488e7 q^{10} +5.40173e8 q^{11} -3.45398e7 q^{13} +1.48918e9 q^{14} +1.55311e10 q^{16} +9.43866e9 q^{17} -2.25061e9 q^{19} +1.85943e11 q^{20} +3.51687e10 q^{22} +3.45854e11 q^{23} +1.38635e12 q^{25} -2.24876e9 q^{26} -2.90106e12 q^{28} -5.11838e11 q^{29} +1.14436e11 q^{31} +3.21203e12 q^{32} +6.14516e11 q^{34} -3.35329e13 q^{35} -1.56972e13 q^{37} -1.46529e11 q^{38} +2.46167e13 q^{40} +8.00761e13 q^{41} -3.66737e13 q^{43} -6.85118e13 q^{44} +2.25173e13 q^{46} +1.17577e14 q^{47} +2.90545e14 q^{49} +9.02599e13 q^{50} +4.38080e12 q^{52} +3.90043e14 q^{53} -7.91917e14 q^{55} -3.84067e14 q^{56} -3.33239e13 q^{58} -1.82562e15 q^{59} +1.41053e15 q^{61} +7.45051e12 q^{62} -1.82656e15 q^{64} +5.06369e13 q^{65} -1.47114e15 q^{67} -1.19713e15 q^{68} -2.18320e15 q^{70} +7.31441e15 q^{71} +1.34580e16 q^{73} -1.02199e15 q^{74} +2.85452e14 q^{76} +1.23554e16 q^{77} -8.37779e15 q^{79} -2.27692e16 q^{80} +5.21346e15 q^{82} +2.55978e16 q^{83} -1.38375e16 q^{85} -2.38769e15 q^{86} -9.07018e15 q^{88} -4.47540e16 q^{89} -7.90031e14 q^{91} -4.38657e16 q^{92} +7.65502e15 q^{94} +3.29949e15 q^{95} +7.41658e16 q^{97} +1.89163e16 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 594 q^{2} + 176516 q^{4} - 382860 q^{5} + 24471568 q^{7} - 130340232 q^{8} - 809382420 q^{10} + 987553512 q^{11} - 2519398244 q^{13} + 435565296 q^{14} + 50611323920 q^{16} + 34313126364 q^{17} + 80053542184 q^{19}+ \cdots + 17\!\cdots\!26 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 65.1063 0.179833 0.0899163 − 0.995949i \(-0.471340\pi\)
0.0899163 + 0.995949i \(0.471340\pi\)
\(3\) 0 0
\(4\) −126833. −0.967660
\(5\) −1.46604e6 −1.67843 −0.839213 − 0.543803i \(-0.816984\pi\)
−0.839213 + 0.543803i \(0.816984\pi\)
\(6\) 0 0
\(7\) 2.28730e7 1.49965 0.749825 − 0.661636i \(-0.230137\pi\)
0.749825 + 0.661636i \(0.230137\pi\)
\(8\) −1.67913e7 −0.353849
\(9\) 0 0
\(10\) −9.54488e7 −0.301836
\(11\) 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 \(0.501869\pi\)
\(14\) 1.48918e9 0.269686
\(15\) 0 0
\(16\) 1.55311e10 0.904027
\(17\) 9.43866e9 0.328167 0.164083 − 0.986446i \(-0.447533\pi\)
0.164083 + 0.986446i \(0.447533\pi\)
\(18\) 0 0
\(19\) −2.25061e9 −0.0304015 −0.0152007 − 0.999884i \(-0.504839\pi\)
−0.0152007 + 0.999884i \(0.504839\pi\)
\(20\) 1.85943e11 1.62415
\(21\) 0 0
\(22\) 3.51687e10 0.136635
\(23\) 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
\(26\) −2.24876e9 −0.00211188
\(27\) 0 0
\(28\) −2.90106e12 −1.45115
\(29\) −5.11838e11 −0.189998 −0.0949992 − 0.995477i \(-0.530285\pi\)
−0.0949992 + 0.995477i \(0.530285\pi\)
\(30\) 0 0
\(31\) 1.14436e11 0.0240984 0.0120492 − 0.999927i \(-0.496165\pi\)
0.0120492 + 0.999927i \(0.496165\pi\)
\(32\) 3.21203e12 0.516423
\(33\) 0 0
\(34\) 6.14516e11 0.0590150
\(35\) −3.35329e13 −2.51705
\(36\) 0 0
\(37\) −1.56972e13 −0.734697 −0.367349 − 0.930083i \(-0.619734\pi\)
−0.367349 + 0.930083i \(0.619734\pi\)
\(38\) −1.46529e11 −0.00546717
\(39\) 0 0
\(40\) 2.46167e13 0.593910
\(41\) 8.00761e13 1.56617 0.783087 − 0.621912i \(-0.213644\pi\)
0.783087 + 0.621912i \(0.213644\pi\)
\(42\) 0 0
\(43\) −3.66737e13 −0.478489 −0.239245 − 0.970959i \(-0.576900\pi\)
−0.239245 + 0.970959i \(0.576900\pi\)
\(44\) −6.85118e13 −0.735221
\(45\) 0 0
\(46\) 2.25173e13 0.165605
\(47\) 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
\(50\) 9.02599e13 0.326776
\(51\) 0 0
\(52\) 4.38080e12 0.0113638
\(53\) 3.90043e14 0.860534 0.430267 − 0.902702i \(-0.358419\pi\)
0.430267 + 0.902702i \(0.358419\pi\)
\(54\) 0 0
\(55\) −7.91917e14 −1.27525
\(56\) −3.84067e14 −0.530651
\(57\) 0 0
\(58\) −3.33239e13 −0.0341679
\(59\) −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 \(0.343883\pi\)
\(62\) 7.45051e12 0.00433368
\(63\) 0 0
\(64\) −1.82656e15 −0.811157
\(65\) 5.06369e13 0.0197108
\(66\) 0 0
\(67\) −1.47114e15 −0.442607 −0.221304 − 0.975205i \(-0.571031\pi\)
−0.221304 + 0.975205i \(0.571031\pi\)
\(68\) −1.19713e15 −0.317554
\(69\) 0 0
\(70\) −2.18320e15 −0.452648
\(71\) 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 \(0.0690302\pi\)
\(74\) −1.02199e15 −0.132122
\(75\) 0 0
\(76\) 2.85452e14 0.0294183
\(77\) 1.23554e16 1.13942
\(78\) 0 0
\(79\) −8.37779e15 −0.621297 −0.310648 − 0.950525i \(-0.600546\pi\)
−0.310648 + 0.950525i \(0.600546\pi\)
\(80\) −2.27692e16 −1.51734
\(81\) 0 0
\(82\) 5.21346e15 0.281649
\(83\) 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
\(86\) −2.38769e15 −0.0860480
\(87\) 0 0
\(88\) −9.07018e15 −0.268852
\(89\) −4.47540e16 −1.20508 −0.602541 − 0.798088i \(-0.705845\pi\)
−0.602541 + 0.798088i \(0.705845\pi\)
\(90\) 0 0
\(91\) −7.90031e14 −0.0176113
\(92\) −4.38657e16 −0.891105
\(93\) 0 0
\(94\) 7.65502e15 0.129527
\(95\) 3.29949e15 0.0510266
\(96\) 0 0
\(97\) 7.41658e16 0.960824 0.480412 − 0.877043i \(-0.340487\pi\)
0.480412 + 0.877043i \(0.340487\pi\)
\(98\) 1.89163e16 0.224602
\(99\) 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 9.18.a.c.1.2 2
3.2 odd 2 3.18.a.b.1.1 ✓ 2
12.11 even 2 48.18.a.h.1.2 2
15.2 even 4 75.18.b.c.49.2 4
15.8 even 4 75.18.b.c.49.3 4
15.14 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 3.2 odd 2
9.18.a.c.1.2 2 1.1 even 1 trivial
48.18.a.h.1.2 2 12.11 even 2
75.18.a.b.1.2 2 15.14 odd 2
75.18.b.c.49.2 4 15.2 even 4
75.18.b.c.49.3 4 15.8 even 4