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Properties

Label 273.2.bd.b.127.7
Level $273$
Weight $2$
Character 273.127
Analytic conductor $2.180$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [273,2,Mod(43,273)] 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("273.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(273, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.bd (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 10 x^{14} - 8 x^{13} - 3 x^{12} + 32 x^{11} - 5 x^{10} - 44 x^{9} + 214 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 127.7
Root \(0.590887 + 1.62814i\) of defining polynomial
Character \(\chi\) \(=\) 273.127
Dual form 273.2.bd.b.43.7

$q$-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+(1.85837 - 1.07293i) q^{2} +(0.500000 + 0.866025i) q^{3} +(1.30235 - 2.25573i) q^{4} -1.91954i q^{5} +(1.85837 + 1.07293i) q^{6} +(0.866025 + 0.500000i) q^{7} -1.29759i q^{8} +(-0.500000 + 0.866025i) q^{9} +(-2.05953 - 3.56721i) q^{10} +(-1.23117 + 0.710818i) q^{11} +2.60470 q^{12} +(1.05481 - 3.44781i) q^{13} +2.14586 q^{14} +(1.66237 - 0.959770i) q^{15} +(1.21247 + 2.10007i) q^{16} +(-3.23553 + 5.60411i) q^{17} +2.14586i q^{18} +(-4.97497 - 2.87230i) q^{19} +(-4.32997 - 2.49991i) q^{20} +1.00000i q^{21} +(-1.52531 + 2.64192i) q^{22} +(-1.07136 - 1.85564i) q^{23} +(1.12375 - 0.648797i) q^{24} +1.31537 q^{25} +(-1.73903 - 7.53902i) q^{26} -1.00000 q^{27} +(2.25573 - 1.30235i) q^{28} +(3.81988 + 6.61623i) q^{29} +(2.05953 - 3.56721i) q^{30} -2.50499i q^{31} +(6.75393 + 3.89939i) q^{32} +(-1.23117 - 0.710818i) q^{33} +13.8860i q^{34} +(0.959770 - 1.66237i) q^{35} +(1.30235 + 2.25573i) q^{36} +(-6.77537 + 3.91176i) q^{37} -12.3271 q^{38} +(3.51329 - 0.810413i) q^{39} -2.49078 q^{40} +(-7.52965 + 4.34725i) q^{41} +(1.07293 + 1.85837i) q^{42} +(3.53722 - 6.12665i) q^{43} +3.70293i q^{44} +(1.66237 + 0.959770i) q^{45} +(-3.98194 - 2.29897i) q^{46} +0.362849i q^{47} +(-1.21247 + 2.10007i) q^{48} +(0.500000 + 0.866025i) q^{49} +(2.44443 - 1.41129i) q^{50} -6.47107 q^{51} +(-6.40361 - 6.86962i) q^{52} +0.524776 q^{53} +(-1.85837 + 1.07293i) q^{54} +(1.36444 + 2.36328i) q^{55} +(0.648797 - 1.12375i) q^{56} -5.74461i q^{57} +(14.1975 + 8.19691i) q^{58} +(-7.80101 - 4.50392i) q^{59} -4.99982i q^{60} +(3.95123 - 6.84374i) q^{61} +(-2.68767 - 4.65518i) q^{62} +(-0.866025 + 0.500000i) q^{63} +11.8851 q^{64} +(-6.61820 - 2.02475i) q^{65} -3.05062 q^{66} +(-0.351053 + 0.202681i) q^{67} +(8.42759 + 14.5970i) q^{68} +(1.07136 - 1.85564i) q^{69} -4.11906i q^{70} +(0.210713 + 0.121655i) q^{71} +(1.12375 + 0.648797i) q^{72} -0.330581i q^{73} +(-8.39408 + 14.5390i) q^{74} +(0.657684 + 1.13914i) q^{75} +(-12.9583 + 7.48148i) q^{76} -1.42164 q^{77} +(5.65947 - 5.27555i) q^{78} +8.64577 q^{79} +(4.03116 - 2.32739i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-9.32857 + 16.1576i) q^{82} -11.4277i q^{83} +(2.25573 + 1.30235i) q^{84} +(10.7573 + 6.21074i) q^{85} -15.1807i q^{86} +(-3.81988 + 6.61623i) q^{87} +(0.922352 + 1.59756i) q^{88} +(10.9755 - 6.33670i) q^{89} +4.11906 q^{90} +(2.63739 - 2.45848i) q^{91} -5.58111 q^{92} +(2.16938 - 1.25249i) q^{93} +(0.389311 + 0.674306i) q^{94} +(-5.51350 + 9.54966i) q^{95} +7.79877i q^{96} +(14.8509 + 8.57418i) q^{97} +(1.85837 + 1.07293i) q^{98} -1.42164i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{3} + 14 q^{4} - 8 q^{9} - 4 q^{10} + 28 q^{12} - 12 q^{13} - 4 q^{14} - 12 q^{15} - 10 q^{16} - 2 q^{17} + 18 q^{20} - 18 q^{22} - 6 q^{23} - 20 q^{25} + 20 q^{26} - 16 q^{27} - 12 q^{29} + 4 q^{30}+ \cdots - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.85837 − 1.07293i 1.31406 − 0.758675i 0.331297 − 0.943527i \(-0.392514\pi\)
0.982766 + 0.184852i \(0.0591806\pi\)
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 1.30235 − 2.25573i 0.651174 − 1.12787i
\(5\) − 1.91954i − 0.858444i −0.903199 − 0.429222i \(-0.858788\pi\)
0.903199 − 0.429222i \(-0.141212\pi\)
\(6\) 1.85837 + 1.07293i 0.758675 + 0.438021i
\(7\) 0.866025 + 0.500000i 0.327327 + 0.188982i
\(8\) − 1.29759i − 0.458768i
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) −2.05953 − 3.56721i −0.651280 − 1.12805i
\(11\) −1.23117 + 0.710818i −0.371212 + 0.214320i −0.673988 − 0.738742i \(-0.735420\pi\)
0.302776 + 0.953062i \(0.402087\pi\)
\(12\) 2.60470 0.751911
\(13\) 1.05481 − 3.44781i 0.292551 − 0.956250i
\(14\) 2.14586 0.573504
\(15\) 1.66237 − 0.959770i 0.429222 − 0.247811i
\(16\) 1.21247 + 2.10007i 0.303118 + 0.525016i
\(17\) −3.23553 + 5.60411i −0.784732 + 1.35920i 0.144426 + 0.989516i \(0.453866\pi\)
−0.929159 + 0.369681i \(0.879467\pi\)
\(18\) 2.14586i 0.505783i
\(19\) −4.97497 − 2.87230i −1.14134 − 0.658952i −0.194576 − 0.980887i \(-0.562333\pi\)
−0.946761 + 0.321936i \(0.895666\pi\)
\(20\) −4.32997 − 2.49991i −0.968211 − 0.558997i
\(21\) 1.00000i 0.218218i
\(22\) −1.52531 + 2.64192i −0.325198 + 0.563259i
\(23\) −1.07136 − 1.85564i −0.223393 − 0.386928i 0.732443 − 0.680828i \(-0.238380\pi\)
−0.955836 + 0.293900i \(0.905047\pi\)
\(24\) 1.12375 − 0.648797i 0.229384 − 0.132435i
\(25\) 1.31537 0.263074
\(26\) −1.73903 − 7.53902i −0.341052 − 1.47852i
\(27\) −1.00000 −0.192450
\(28\) 2.25573 − 1.30235i 0.426294 − 0.246121i
\(29\) 3.81988 + 6.61623i 0.709334 + 1.22860i 0.965104 + 0.261865i \(0.0843376\pi\)
−0.255770 + 0.966738i \(0.582329\pi\)
\(30\) 2.05953 − 3.56721i 0.376017 − 0.651280i
\(31\) − 2.50499i − 0.449909i −0.974369 − 0.224954i \(-0.927777\pi\)
0.974369 − 0.224954i \(-0.0722233\pi\)
\(32\) 6.75393 + 3.89939i 1.19394 + 0.689321i
\(33\) −1.23117 − 0.710818i −0.214320 − 0.123737i
\(34\) 13.8860i 2.38143i
\(35\) 0.959770 − 1.66237i 0.162231 − 0.280992i
\(36\) 1.30235 + 2.25573i 0.217058 + 0.375956i
\(37\) −6.77537 + 3.91176i −1.11386 + 0.643090i −0.939827 − 0.341649i \(-0.889014\pi\)
−0.174037 + 0.984739i \(0.555681\pi\)
\(38\) −12.3271 −1.99972
\(39\) 3.51329 − 0.810413i 0.562577 − 0.129770i
\(40\) −2.49078 −0.393827
\(41\) −7.52965 + 4.34725i −1.17593 + 0.678926i −0.955071 − 0.296379i \(-0.904221\pi\)
−0.220864 + 0.975305i \(0.570888\pi\)
\(42\) 1.07293 + 1.85837i 0.165556 + 0.286752i
\(43\) 3.53722 − 6.12665i 0.539422 − 0.934306i −0.459514 − 0.888171i \(-0.651976\pi\)
0.998935 − 0.0461349i \(-0.0146904\pi\)
\(44\) 3.70293i 0.558238i
\(45\) 1.66237 + 0.959770i 0.247811 + 0.143074i
\(46\) −3.98194 − 2.29897i −0.587105 − 0.338965i
\(47\) 0.362849i 0.0529270i 0.999650 + 0.0264635i \(0.00842457\pi\)
−0.999650 + 0.0264635i \(0.991575\pi\)
\(48\) −1.21247 + 2.10007i −0.175005 + 0.303118i
\(49\) 0.500000 + 0.866025i 0.0714286 + 0.123718i
\(50\) 2.44443 − 1.41129i 0.345695 − 0.199587i
\(51\) −6.47107 −0.906131
\(52\) −6.40361 − 6.86962i −0.888021 − 0.952644i
\(53\) 0.524776 0.0720835 0.0360417 − 0.999350i \(-0.488525\pi\)
0.0360417 + 0.999350i \(0.488525\pi\)
\(54\) −1.85837 + 1.07293i −0.252892 + 0.146007i
\(55\) 1.36444 + 2.36328i 0.183981 + 0.318665i
\(56\) 0.648797 − 1.12375i 0.0866991 − 0.150167i
\(57\) − 5.74461i − 0.760892i
\(58\) 14.1975 + 8.19691i 1.86422 + 1.07631i
\(59\) −7.80101 − 4.50392i −1.01560 − 0.586360i −0.102777 − 0.994704i \(-0.532773\pi\)
−0.912828 + 0.408345i \(0.866106\pi\)
\(60\) − 4.99982i − 0.645474i
\(61\) 3.95123 − 6.84374i 0.505904 − 0.876251i −0.494073 − 0.869420i \(-0.664492\pi\)
0.999977 − 0.00683058i \(-0.00217426\pi\)
\(62\) −2.68767 − 4.65518i −0.341334 − 0.591208i
\(63\) −0.866025 + 0.500000i −0.109109 + 0.0629941i
\(64\) 11.8851 1.48564
\(65\) −6.61820 − 2.02475i −0.820887 − 0.251139i
\(66\) −3.05062 −0.375506
\(67\) −0.351053 + 0.202681i −0.0428879 + 0.0247614i −0.521291 − 0.853379i \(-0.674549\pi\)
0.478403 + 0.878141i \(0.341216\pi\)
\(68\) 8.42759 + 14.5970i 1.02200 + 1.77015i
\(69\) 1.07136 − 1.85564i 0.128976 − 0.223393i
\(70\) − 4.11906i − 0.492321i
\(71\) 0.210713 + 0.121655i 0.0250071 + 0.0144378i 0.512451 − 0.858716i \(-0.328737\pi\)
−0.487444 + 0.873154i \(0.662071\pi\)
\(72\) 1.12375 + 0.648797i 0.132435 + 0.0764614i
\(73\) − 0.330581i − 0.0386916i −0.999813 − 0.0193458i \(-0.993842\pi\)
0.999813 − 0.0193458i \(-0.00615835\pi\)
\(74\) −8.39408 + 14.5390i −0.975792 + 1.69012i
\(75\) 0.657684 + 1.13914i 0.0759428 + 0.131537i
\(76\) −12.9583 + 7.48148i −1.48642 + 0.858185i
\(77\) −1.42164 −0.162010
\(78\) 5.65947 − 5.27555i 0.640809 − 0.597339i
\(79\) 8.64577 0.972725 0.486363 − 0.873757i \(-0.338323\pi\)
0.486363 + 0.873757i \(0.338323\pi\)
\(80\) 4.03116 − 2.32739i 0.450697 − 0.260210i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −9.32857 + 16.1576i −1.03017 + 1.78430i
\(83\) − 11.4277i − 1.25435i −0.778877 − 0.627177i \(-0.784210\pi\)
0.778877 − 0.627177i \(-0.215790\pi\)
\(84\) 2.25573 + 1.30235i 0.246121 + 0.142098i
\(85\) 10.7573 + 6.21074i 1.16679 + 0.673649i
\(86\) − 15.1807i − 1.63698i
\(87\) −3.81988 + 6.61623i −0.409534 + 0.709334i
\(88\) 0.922352 + 1.59756i 0.0983231 + 0.170301i
\(89\) 10.9755 − 6.33670i 1.16340 − 0.671689i 0.211284 − 0.977425i \(-0.432236\pi\)
0.952117 + 0.305735i \(0.0989023\pi\)
\(90\) 4.11906 0.434187
\(91\) 2.63739 − 2.45848i 0.276474 − 0.257719i
\(92\) −5.58111 −0.581871
\(93\) 2.16938 − 1.25249i 0.224954 − 0.129877i
\(94\) 0.389311 + 0.674306i 0.0401543 + 0.0695494i
\(95\) −5.51350 + 9.54966i −0.565673 + 0.979775i
\(96\) 7.79877i 0.795959i
\(97\) 14.8509 + 8.57418i 1.50788 + 0.870576i 0.999958 + 0.00917340i \(0.00292002\pi\)
0.507923 + 0.861402i \(0.330413\pi\)
\(98\) 1.85837 + 1.07293i 0.187723 + 0.108382i
\(99\) − 1.42164i − 0.142880i
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 273.2.bd.b.127.7 yes 16
3.2 odd 2 819.2.ct.c.127.2 16
13.2 odd 12 3549.2.a.ba.1.7 8
13.4 even 6 inner 273.2.bd.b.43.7 ✓ 16
13.11 odd 12 3549.2.a.bc.1.2 8
39.17 odd 6 819.2.ct.c.316.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.bd.b.43.7 ✓ 16 13.4 even 6 inner
273.2.bd.b.127.7 yes 16 1.1 even 1 trivial
819.2.ct.c.127.2 16 3.2 odd 2
819.2.ct.c.316.2 16 39.17 odd 6
3549.2.a.ba.1.7 8 13.2 odd 12
3549.2.a.bc.1.2 8 13.11 odd 12