Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,26,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, 26); 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, 26, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(35.6397101957\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 783420x + 148321440 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{6}\cdot 3^{5}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(768.963\) 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)
 
\(f(q)\) \(=\) \(q-8744.24 q^{2} +4.29073e7 q^{4} -4.98342e7 q^{5} +5.50645e10 q^{7} -8.17835e10 q^{8} +4.35762e11 q^{10} +1.39744e13 q^{11} +1.27551e14 q^{13} -4.81498e14 q^{14} -7.24595e14 q^{16} -1.41846e15 q^{17} -1.02899e16 q^{19} -2.13825e15 q^{20} -1.22196e17 q^{22} +6.75984e16 q^{23} -2.95540e17 q^{25} -1.11533e18 q^{26} +2.36267e18 q^{28} +3.07242e18 q^{29} +9.03157e17 q^{31} +9.08023e18 q^{32} +1.24034e19 q^{34} -2.74410e18 q^{35} -5.36810e18 q^{37} +8.99775e19 q^{38} +4.07562e18 q^{40} -1.40813e20 q^{41} +1.70911e20 q^{43} +5.99604e20 q^{44} -5.91097e20 q^{46} +6.32772e20 q^{47} +1.69104e21 q^{49} +2.58427e21 q^{50} +5.47286e21 q^{52} -9.60635e20 q^{53} -6.96404e20 q^{55} -4.50337e21 q^{56} -2.68659e22 q^{58} -1.44777e22 q^{59} +3.87814e21 q^{61} -7.89742e21 q^{62} -5.50863e22 q^{64} -6.35639e21 q^{65} +8.52923e22 q^{67} -6.08624e22 q^{68} +2.39951e22 q^{70} -7.92209e22 q^{71} +2.67527e23 q^{73} +4.69400e22 q^{74} -4.41512e23 q^{76} +7.69494e23 q^{77} -6.21605e23 q^{79} +3.61096e22 q^{80} +1.23130e24 q^{82} -3.18211e23 q^{83} +7.06881e22 q^{85} -1.49449e24 q^{86} -1.14288e24 q^{88} +3.79033e24 q^{89} +7.02353e24 q^{91} +2.90047e24 q^{92} -5.53311e24 q^{94} +5.12790e23 q^{95} +1.55884e24 q^{97} -1.47868e25 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3678 q^{2} + 90747588 q^{4} + 163152750 q^{5} - 9622572744 q^{7} + 268012830648 q^{8} + 8363188874700 q^{10} + 5946998130780 q^{11} + 248137774407690 q^{13} - 754361641264848 q^{14} + 23\!\cdots\!16 q^{16}+ \cdots - 24\!\cdots\!50 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8744.24 −1.50955 −0.754774 − 0.655985i \(-0.772254\pi\)
−0.754774 + 0.655985i \(0.772254\pi\)
\(3\) 0 0
\(4\) 4.29073e7 1.27874
\(5\) −4.98342e7 −0.0912857 −0.0456428 − 0.998958i \(-0.514534\pi\)
−0.0456428 + 0.998958i \(0.514534\pi\)
\(6\) 0 0
\(7\) 5.50645e10 1.50365 0.751825 − 0.659363i \(-0.229174\pi\)
0.751825 + 0.659363i \(0.229174\pi\)
\(8\) −8.17835e10 −0.420766
\(9\) 0 0
\(10\) 4.35762e11 0.137800
\(11\) 1.39744e13 1.34253 0.671266 − 0.741216i \(-0.265751\pi\)
0.671266 + 0.741216i \(0.265751\pi\)
\(12\) 0 0
\(13\) 1.27551e14 1.51842 0.759209 − 0.650847i \(-0.225586\pi\)
0.759209 + 0.650847i \(0.225586\pi\)
\(14\) −4.81498e14 −2.26983
\(15\) 0 0
\(16\) −7.24595e14 −0.643569
\(17\) −1.41846e15 −0.590482 −0.295241 − 0.955423i \(-0.595400\pi\)
−0.295241 + 0.955423i \(0.595400\pi\)
\(18\) 0 0
\(19\) −1.02899e16 −1.06658 −0.533288 − 0.845934i \(-0.679044\pi\)
−0.533288 + 0.845934i \(0.679044\pi\)
\(20\) −2.13825e15 −0.116730
\(21\) 0 0
\(22\) −1.22196e17 −2.02662
\(23\) 6.75984e16 0.643189 0.321595 − 0.946877i \(-0.395781\pi\)
0.321595 + 0.946877i \(0.395781\pi\)
\(24\) 0 0
\(25\) −2.95540e17 −0.991667
\(26\) −1.11533e18 −2.29212
\(27\) 0 0
\(28\) 2.36267e18 1.92277
\(29\) 3.07242e18 1.61252 0.806260 − 0.591562i \(-0.201488\pi\)
0.806260 + 0.591562i \(0.201488\pi\)
\(30\) 0 0
\(31\) 9.03157e17 0.205941 0.102970 − 0.994684i \(-0.467165\pi\)
0.102970 + 0.994684i \(0.467165\pi\)
\(32\) 9.08023e18 1.39227
\(33\) 0 0
\(34\) 1.24034e19 0.891362
\(35\) −2.74410e18 −0.137262
\(36\) 0 0
\(37\) −5.36810e18 −0.134060 −0.0670300 − 0.997751i \(-0.521352\pi\)
−0.0670300 + 0.997751i \(0.521352\pi\)
\(38\) 8.99775e19 1.61005
\(39\) 0 0
\(40\) 4.07562e18 0.0384100
\(41\) −1.40813e20 −0.974640 −0.487320 − 0.873223i \(-0.662025\pi\)
−0.487320 + 0.873223i \(0.662025\pi\)
\(42\) 0 0
\(43\) 1.70911e20 0.652250 0.326125 − 0.945327i \(-0.394257\pi\)
0.326125 + 0.945327i \(0.394257\pi\)
\(44\) 5.99604e20 1.71675
\(45\) 0 0
\(46\) −5.91097e20 −0.970925
\(47\) 6.32772e20 0.794372 0.397186 − 0.917738i \(-0.369987\pi\)
0.397186 + 0.917738i \(0.369987\pi\)
\(48\) 0 0
\(49\) 1.69104e21 1.26096
\(50\) 2.58427e21 1.49697
\(51\) 0 0
\(52\) 5.47286e21 1.94166
\(53\) −9.60635e20 −0.268602 −0.134301 − 0.990941i \(-0.542879\pi\)
−0.134301 + 0.990941i \(0.542879\pi\)
\(54\) 0 0
\(55\) −6.96404e20 −0.122554
\(56\) −4.50337e21 −0.632685
\(57\) 0 0
\(58\) −2.68659e22 −2.43418
\(59\) −1.44777e22 −1.05937 −0.529687 − 0.848193i \(-0.677691\pi\)
−0.529687 + 0.848193i \(0.677691\pi\)
\(60\) 0 0
\(61\) 3.87814e21 0.187069 0.0935343 − 0.995616i \(-0.470184\pi\)
0.0935343 + 0.995616i \(0.470184\pi\)
\(62\) −7.89742e21 −0.310877
\(63\) 0 0
\(64\) −5.50863e22 −1.45812
\(65\) −6.35639e21 −0.138610
\(66\) 0 0
\(67\) 8.52923e22 1.27343 0.636715 − 0.771100i \(-0.280293\pi\)
0.636715 + 0.771100i \(0.280293\pi\)
\(68\) −6.08624e22 −0.755071
\(69\) 0 0
\(70\) 2.39951e22 0.207203
\(71\) −7.92209e22 −0.572941 −0.286471 − 0.958089i \(-0.592482\pi\)
−0.286471 + 0.958089i \(0.592482\pi\)
\(72\) 0 0
\(73\) 2.67527e23 1.36720 0.683599 − 0.729858i \(-0.260414\pi\)
0.683599 + 0.729858i \(0.260414\pi\)
\(74\) 4.69400e22 0.202370
\(75\) 0 0
\(76\) −4.41512e23 −1.36387
\(77\) 7.69494e23 2.01870
\(78\) 0 0
\(79\) −6.21605e23 −1.18352 −0.591760 − 0.806114i \(-0.701567\pi\)
−0.591760 + 0.806114i \(0.701567\pi\)
\(80\) 3.61096e22 0.0587487
\(81\) 0 0
\(82\) 1.23130e24 1.47127
\(83\) −3.18211e23 −0.326767 −0.163384 − 0.986563i \(-0.552241\pi\)
−0.163384 + 0.986563i \(0.552241\pi\)
\(84\) 0 0
\(85\) 7.06881e22 0.0539026
\(86\) −1.49449e24 −0.984603
\(87\) 0 0
\(88\) −1.14288e24 −0.564893
\(89\) 3.79033e24 1.62668 0.813339 − 0.581790i \(-0.197648\pi\)
0.813339 + 0.581790i \(0.197648\pi\)
\(90\) 0 0
\(91\) 7.02353e24 2.28317
\(92\) 2.90047e24 0.822470
\(93\) 0 0
\(94\) −5.53311e24 −1.19914
\(95\) 5.12790e23 0.0973631
\(96\) 0 0
\(97\) 1.55884e24 0.228115 0.114058 − 0.993474i \(-0.463615\pi\)
0.114058 + 0.993474i \(0.463615\pi\)
\(98\) −1.47868e25 −1.90348
\(99\) 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 9.26.a.c.1.1 3
3.2 odd 2 3.26.a.b.1.3 ✓ 3
12.11 even 2 48.26.a.i.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.26.a.b.1.3 ✓ 3 3.2 odd 2
9.26.a.c.1.1 3 1.1 even 1 trivial
48.26.a.i.1.2 3 12.11 even 2