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 361.1
Root \(-0.984808 + 0.173648i\) of defining polynomial
Character \(\chi\) \(=\) 666.361
Dual form 666.2.bj.c.559.1

$q$-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 \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(e\left(\frac{17}{18}\right)\)

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 −0.707903 − 0.706310i \(-0.750359\pi\)
0.818506 + 0.574499i \(0.194803\pi\)
\(6\) 0 0
\(7\) −2.50048 − 2.09815i −0.945091 − 0.793026i 0.0333729 − 0.999443i \(-0.489375\pi\)
−0.978464 + 0.206417i \(0.933820\pi\)
\(8\) −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 0.992404 − 0.123023i \(-0.0392590\pi\)
−0.602743 + 0.797935i \(0.705926\pi\)
\(12\) 0 0
\(13\) −0.466831 − 1.28261i −0.129476 − 0.355731i 0.857968 − 0.513703i \(-0.171727\pi\)
−0.987444 + 0.157972i \(0.949504\pi\)
\(14\) 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 −0.721356 − 0.692565i \(-0.756481\pi\)
0.997763 − 0.0668568i \(-0.0212971\pi\)
\(18\) 0 0
\(19\) −5.89485 − 1.03942i −1.35237 − 0.238459i −0.549939 − 0.835205i \(-0.685349\pi\)
−0.802431 + 0.596745i \(0.796460\pi\)
\(20\) 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 −0.372680 − 0.927960i \(-0.621561\pi\)
−0.989977 + 0.141230i \(0.954894\pi\)
\(24\) 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 −0.735583 − 0.677434i \(-0.763092\pi\)
0.218883 + 0.975751i \(0.429759\pi\)
\(30\) 0 0
\(31\) − 2.53737i − 0.455726i −0.973693 − 0.227863i \(-0.926826\pi\)
0.973693 − 0.227863i \(-0.0731738\pi\)
\(32\) −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 −0.867931 − 0.496685i \(-0.834551\pi\)
−0.345611 + 0.938378i \(0.612328\pi\)
\(42\) 0 0
\(43\) 4.33920i 0.661722i 0.943680 + 0.330861i \(0.107339\pi\)
−0.943680 + 0.330861i \(0.892661\pi\)
\(44\) 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 −0.609888 − 0.792488i \(-0.708785\pi\)
0.991258 + 0.131934i \(0.0421188\pi\)
\(48\) 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 −0.0601551 − 0.998189i \(-0.519160\pi\)
0.972578 + 0.232575i \(0.0747151\pi\)
\(54\) 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 −0.953315 − 0.301978i \(-0.902353\pi\)
−0.131849 − 0.991270i \(-0.542091\pi\)
\(60\) 0 0
\(61\) −2.39847 − 6.58973i −0.307092 − 0.843728i −0.993220 − 0.116248i \(-0.962913\pi\)
0.686128 − 0.727481i \(-0.259309\pi\)
\(62\) 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 −0.0584586 − 0.998290i \(-0.518619\pi\)
−0.993275 + 0.115781i \(0.963063\pi\)
\(68\) − 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 0.294363 + 0.955694i \(0.404893\pi\)
−0.603477 + 0.797380i \(0.706219\pi\)
\(72\) 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 −0.710532 − 0.703664i \(-0.751546\pi\)
0.816357 + 0.577548i \(0.195990\pi\)
\(80\) − 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 0.341447 − 0.939901i \(-0.389083\pi\)
−0.342593 + 0.939484i \(0.611305\pi\)
\(84\) 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 0.797542 − 0.603264i \(-0.206133\pi\)
−0.732591 + 0.680669i \(0.761689\pi\)
\(90\) 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 0.980920 − 0.194410i \(-0.0622791\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(98\) −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.361.1 12
3.2 odd 2 74.2.h.a.65.2 yes 12
12.11 even 2 592.2.bq.b.65.2 12
37.4 even 18 inner 666.2.bj.c.559.1 12
111.2 even 36 2738.2.a.s.1.6 6
111.35 even 36 2738.2.a.r.1.5 6
111.41 odd 18 74.2.h.a.41.2 ✓ 12
444.263 even 18 592.2.bq.b.337.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.41.2 ✓ 12 111.41 odd 18
74.2.h.a.65.2 yes 12 3.2 odd 2
592.2.bq.b.65.2 12 12.11 even 2
592.2.bq.b.337.2 12 444.263 even 18
666.2.bj.c.361.1 12 1.1 even 1 trivial
666.2.bj.c.559.1 12 37.4 even 18 inner
2738.2.a.r.1.5 6 111.35 even 36
2738.2.a.s.1.6 6 111.2 even 36