Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [666,2,Mod(289,666)] 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("666.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(666, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.bj (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,-12,0,0,6,6,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 559.1
Root \(-0.984808 - 0.173648i\) of defining polynomial
Character \(\chi\) \(=\) 666.559
Dual form 666.2.bj.c.361.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.984808 - 0.173648i) q^{2} +(0.939693 + 0.342020i) q^{4} +(0.247315 + 0.294739i) q^{5} +(-2.50048 + 2.09815i) q^{7} +(-0.866025 - 0.500000i) q^{8} +(-0.192377 - 0.333207i) q^{10} +(1.29236 - 2.23843i) q^{11} +(-0.466831 + 1.28261i) q^{13} +(2.82683 - 1.63207i) q^{14} +(0.766044 + 0.642788i) q^{16} +(1.13965 + 3.13118i) q^{17} +(-5.89485 + 1.03942i) q^{19} +(0.131594 + 0.361551i) q^{20} +(-1.66142 + 1.98001i) q^{22} +(-6.53507 + 3.77303i) q^{23} +(0.842535 - 4.77825i) q^{25} +(0.682461 - 1.18206i) q^{26} +(-3.06729 + 1.11640i) q^{28} +(-2.78251 - 1.60649i) q^{29} +2.53737i q^{31} +(-0.642788 - 0.766044i) q^{32} +(-0.578618 - 3.28150i) q^{34} +(-1.23681 - 0.218083i) q^{35} +(0.543196 + 6.05846i) q^{37} +5.98578 q^{38} +(-0.0668119 - 0.378909i) q^{40} +(-7.77046 - 2.82822i) q^{41} -4.33920i q^{43} +(1.98001 - 1.66142i) q^{44} +(7.09097 - 2.58090i) q^{46} +(2.61455 + 4.52853i) q^{47} +(0.634616 - 3.59909i) q^{49} +(-1.65947 + 4.55935i) q^{50} +(-0.877355 + 1.04559i) q^{52} +(6.64254 + 5.57375i) q^{53} +(0.979373 - 0.172690i) q^{55} +(3.21455 - 0.566812i) q^{56} +(2.46128 + 2.06526i) q^{58} +(-8.33530 + 9.93362i) q^{59} +(-2.39847 + 6.58973i) q^{61} +(0.440610 - 2.49882i) q^{62} +(0.500000 + 0.866025i) q^{64} +(-0.493489 + 0.179615i) q^{65} +(-8.60881 + 7.22365i) q^{67} +3.33213i q^{68} +(1.18015 + 0.429540i) q^{70} +(-2.60464 - 14.7717i) q^{71} -15.0792 q^{73} +(0.517097 - 6.06074i) q^{74} +(-5.89485 - 1.03942i) q^{76} +(1.46505 + 8.30870i) q^{77} +(0.940587 + 1.12095i) q^{79} +0.384754i q^{80} +(7.16130 + 4.13458i) q^{82} +(-0.0104473 + 0.00380252i) q^{83} +(-0.641025 + 1.11029i) q^{85} +(-0.753494 + 4.27328i) q^{86} +(-2.23843 + 1.29236i) q^{88} +(0.612745 - 0.730241i) q^{89} +(-1.52380 - 4.18661i) q^{91} +(-7.43141 + 1.31036i) q^{92} +(-1.78845 - 4.91374i) q^{94} +(-1.76424 - 1.48038i) q^{95} +(12.8332 - 7.40927i) q^{97} +(-1.24995 + 3.43421i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{7} + 6 q^{10} + 6 q^{11} + 6 q^{13} + 18 q^{14} - 18 q^{19} - 18 q^{25} - 12 q^{26} - 6 q^{28} - 18 q^{29} + 12 q^{34} - 18 q^{35} + 30 q^{37} + 24 q^{38} + 12 q^{40} - 24 q^{41} - 6 q^{44}+ \cdots - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.984808 − 0.173648i −0.696364 − 0.122788i
\(3\) 0 0
\(4\) 0.939693 + 0.342020i 0.469846 + 0.171010i
\(5\) 0.247315 + 0.294739i 0.110603 + 0.131811i 0.818506 − 0.574499i \(-0.194803\pi\)
−0.707903 + 0.706310i \(0.750359\pi\)
\(6\) 0 0
\(7\) −2.50048 + 2.09815i −0.945091 + 0.793026i −0.978464 − 0.206417i \(-0.933820\pi\)
0.0333729 + 0.999443i \(0.489375\pi\)
\(8\) −0.866025 − 0.500000i −0.306186 − 0.176777i
\(9\) 0 0
\(10\) −0.192377 − 0.333207i −0.0608350 − 0.105369i
\(11\) 1.29236 − 2.23843i 0.389661 − 0.674912i −0.602743 − 0.797935i \(-0.705926\pi\)
0.992404 + 0.123023i \(0.0392590\pi\)
\(12\) 0 0
\(13\) −0.466831 + 1.28261i −0.129476 + 0.355731i −0.987444 − 0.157972i \(-0.949504\pi\)
0.857968 + 0.513703i \(0.171727\pi\)
\(14\) 2.82683 − 1.63207i 0.755502 − 0.436189i
\(15\) 0 0
\(16\) 0.766044 + 0.642788i 0.191511 + 0.160697i
\(17\) 1.13965 + 3.13118i 0.276407 + 0.759422i 0.997763 + 0.0668568i \(0.0212971\pi\)
−0.721356 + 0.692565i \(0.756481\pi\)
\(18\) 0 0
\(19\) −5.89485 + 1.03942i −1.35237 + 0.238459i −0.802431 − 0.596745i \(-0.796460\pi\)
−0.549939 + 0.835205i \(0.685349\pi\)
\(20\) 0.131594 + 0.361551i 0.0294253 + 0.0808452i
\(21\) 0 0
\(22\) −1.66142 + 1.98001i −0.354217 + 0.422139i
\(23\) −6.53507 + 3.77303i −1.36266 + 0.786730i −0.989977 − 0.141230i \(-0.954894\pi\)
−0.372680 + 0.927960i \(0.621561\pi\)
\(24\) 0 0
\(25\) 0.842535 − 4.77825i 0.168507 − 0.955650i
\(26\) 0.682461 − 1.18206i 0.133842 − 0.231821i
\(27\) 0 0
\(28\) −3.06729 + 1.11640i −0.579663 + 0.210980i
\(29\) −2.78251 − 1.60649i −0.516700 − 0.298317i 0.218883 − 0.975751i \(-0.429759\pi\)
−0.735583 + 0.677434i \(0.763092\pi\)
\(30\) 0 0
\(31\) 2.53737i 0.455726i 0.973693 + 0.227863i \(0.0731738\pi\)
−0.973693 + 0.227863i \(0.926826\pi\)
\(32\) −0.642788 − 0.766044i −0.113630 − 0.135419i
\(33\) 0 0
\(34\) −0.578618 − 3.28150i −0.0992321 − 0.562773i
\(35\) −1.23681 − 0.218083i −0.209059 − 0.0368628i
\(36\) 0 0
\(37\) 0.543196 + 6.05846i 0.0893009 + 0.996005i
\(38\) 5.98578 0.971022
\(39\) 0 0
\(40\) −0.0668119 − 0.378909i −0.0105639 − 0.0599108i
\(41\) −7.77046 − 2.82822i −1.21354 − 0.441693i −0.345611 − 0.938378i \(-0.612328\pi\)
−0.867931 + 0.496685i \(0.834551\pi\)
\(42\) 0 0
\(43\) − 4.33920i − 0.661722i −0.943680 − 0.330861i \(-0.892661\pi\)
0.943680 − 0.330861i \(-0.107339\pi\)
\(44\) 1.98001 − 1.66142i 0.298497 − 0.250469i
\(45\) 0 0
\(46\) 7.09097 − 2.58090i 1.04551 − 0.380533i
\(47\) 2.61455 + 4.52853i 0.381371 + 0.660554i 0.991258 − 0.131934i \(-0.0421188\pi\)
−0.609888 + 0.792488i \(0.708785\pi\)
\(48\) 0 0
\(49\) 0.634616 − 3.59909i 0.0906594 − 0.514155i
\(50\) −1.65947 + 4.55935i −0.234684 + 0.644790i
\(51\) 0 0
\(52\) −0.877355 + 1.04559i −0.121667 + 0.144997i
\(53\) 6.64254 + 5.57375i 0.912423 + 0.765614i 0.972578 − 0.232575i \(-0.0747151\pi\)
−0.0601551 + 0.998189i \(0.519160\pi\)
\(54\) 0 0
\(55\) 0.979373 − 0.172690i 0.132059 − 0.0232855i
\(56\) 3.21455 − 0.566812i 0.429562 − 0.0757434i
\(57\) 0 0
\(58\) 2.46128 + 2.06526i 0.323182 + 0.271182i
\(59\) −8.33530 + 9.93362i −1.08516 + 1.29325i −0.131849 + 0.991270i \(0.542091\pi\)
−0.953315 + 0.301978i \(0.902353\pi\)
\(60\) 0 0
\(61\) −2.39847 + 6.58973i −0.307092 + 0.843728i 0.686128 + 0.727481i \(0.259309\pi\)
−0.993220 + 0.116248i \(0.962913\pi\)
\(62\) 0.440610 − 2.49882i 0.0559576 − 0.317351i
\(63\) 0 0
\(64\) 0.500000 + 0.866025i 0.0625000 + 0.108253i
\(65\) −0.493489 + 0.179615i −0.0612098 + 0.0222785i
\(66\) 0 0
\(67\) −8.60881 + 7.22365i −1.05173 + 0.882509i −0.993275 − 0.115781i \(-0.963063\pi\)
−0.0584586 + 0.998290i \(0.518619\pi\)
\(68\) 3.33213i 0.404080i
\(69\) 0 0
\(70\) 1.18015 + 0.429540i 0.141055 + 0.0513399i
\(71\) −2.60464 − 14.7717i −0.309114 − 1.75307i −0.603477 − 0.797380i \(-0.706219\pi\)
0.294363 − 0.955694i \(-0.404893\pi\)
\(72\) 0 0
\(73\) −15.0792 −1.76489 −0.882445 − 0.470416i \(-0.844104\pi\)
−0.882445 + 0.470416i \(0.844104\pi\)
\(74\) 0.517097 − 6.06074i 0.0601113 − 0.704547i
\(75\) 0 0
\(76\) −5.89485 − 1.03942i −0.676185 − 0.119230i
\(77\) 1.46505 + 8.30870i 0.166958 + 0.946864i
\(78\) 0 0
\(79\) 0.940587 + 1.12095i 0.105824 + 0.126117i 0.816357 − 0.577548i \(-0.195990\pi\)
−0.710532 + 0.703664i \(0.751546\pi\)
\(80\) 0.384754i 0.0430169i
\(81\) 0 0
\(82\) 7.16130 + 4.13458i 0.790833 + 0.456588i
\(83\) −0.0104473 + 0.00380252i −0.00114674 + 0.000417381i −0.342593 − 0.939484i \(-0.611305\pi\)
0.341447 + 0.939901i \(0.389083\pi\)
\(84\) 0 0
\(85\) −0.641025 + 1.11029i −0.0695290 + 0.120428i
\(86\) −0.753494 + 4.27328i −0.0812514 + 0.460799i
\(87\) 0 0
\(88\) −2.23843 + 1.29236i −0.238617 + 0.137766i
\(89\) 0.612745 − 0.730241i 0.0649509 − 0.0774054i −0.732591 − 0.680669i \(-0.761689\pi\)
0.797542 + 0.603264i \(0.206133\pi\)
\(90\) 0 0
\(91\) −1.52380 − 4.18661i −0.159738 − 0.438876i
\(92\) −7.43141 + 1.31036i −0.774778 + 0.136614i
\(93\) 0 0
\(94\) −1.78845 − 4.91374i −0.184465 − 0.506814i
\(95\) −1.76424 − 1.48038i −0.181008 − 0.151883i
\(96\) 0 0
\(97\) 12.8332 − 7.40927i 1.30302 − 0.752297i 0.322097 − 0.946707i \(-0.395612\pi\)
0.980920 + 0.194410i \(0.0622791\pi\)
\(98\) −1.24995 + 3.43421i −0.126264 + 0.346907i
\(99\) 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 666.2.bj.c.559.1 12
3.2 odd 2 74.2.h.a.41.2 ✓ 12
12.11 even 2 592.2.bq.b.337.2 12
37.28 even 18 inner 666.2.bj.c.361.1 12
111.56 even 36 2738.2.a.s.1.6 6
111.65 odd 18 74.2.h.a.65.2 yes 12
111.92 even 36 2738.2.a.r.1.5 6
444.287 even 18 592.2.bq.b.65.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.41.2 ✓ 12 3.2 odd 2
74.2.h.a.65.2 yes 12 111.65 odd 18
592.2.bq.b.65.2 12 444.287 even 18
592.2.bq.b.337.2 12 12.11 even 2
666.2.bj.c.361.1 12 37.28 even 18 inner
666.2.bj.c.559.1 12 1.1 even 1 trivial
2738.2.a.r.1.5 6 111.92 even 36
2738.2.a.s.1.6 6 111.56 even 36