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Properties

Label 322.2.e.a.277.4
Level $322$
Weight $2$
Character 322.277
Analytic conductor $2.571$
Analytic rank $0$
Dimension $8$
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] = [322,2,Mod(93,322)] 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("322.93"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(322, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-4,-3,-4,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.310217769.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 277.4
Root \(-0.198169 + 0.343239i\) of defining polynomial
Character \(\chi\) \(=\) 322.277
Dual form 322.2.e.a.93.4

$q$-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+(-0.500000 - 0.866025i) q^{2} +(0.563379 - 0.975800i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-0.858079 - 1.48624i) q^{5} -1.12676 q^{6} +(0.779537 - 2.52830i) q^{7} +1.00000 q^{8} +(0.865209 + 1.49859i) q^{9} +(-0.858079 + 1.48624i) q^{10} +(1.82493 - 3.16087i) q^{11} +(0.563379 + 0.975800i) q^{12} -3.79833 q^{13} +(-2.57934 + 0.589053i) q^{14} -1.93369 q^{15} +(-0.500000 - 0.866025i) q^{16} +(-2.08850 + 3.61738i) q^{17} +(0.865209 - 1.49859i) q^{18} +(-4.06135 - 7.03447i) q^{19} +1.71616 q^{20} +(-2.02795 - 2.18506i) q^{21} -3.64985 q^{22} +(0.500000 + 0.866025i) q^{23} +(0.563379 - 0.975800i) q^{24} +(1.02740 - 1.77951i) q^{25} +(1.89916 + 3.28945i) q^{26} +5.33003 q^{27} +(1.79981 + 1.93925i) q^{28} +2.80694 q^{29} +(0.966847 + 1.67463i) q^{30} +(-4.27150 + 7.39846i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(-2.05625 - 3.56153i) q^{33} +4.17700 q^{34} +(-4.42656 + 1.01091i) q^{35} -1.73042 q^{36} +(-4.46765 - 7.73820i) q^{37} +(-4.06135 + 7.03447i) q^{38} +(-2.13990 + 3.70641i) q^{39} +(-0.858079 - 1.48624i) q^{40} +10.0518 q^{41} +(-0.878349 + 2.84878i) q^{42} +8.11069 q^{43} +(1.82493 + 3.16087i) q^{44} +(1.48484 - 2.57181i) q^{45} +(0.500000 - 0.866025i) q^{46} +(1.47568 + 2.55596i) q^{47} -1.12676 q^{48} +(-5.78464 - 3.94181i) q^{49} -2.05480 q^{50} +(2.35323 + 4.07591i) q^{51} +(1.89916 - 3.28945i) q^{52} +(-1.52740 + 2.64553i) q^{53} +(-2.66502 - 4.61594i) q^{54} -6.26373 q^{55} +(0.779537 - 2.52830i) q^{56} -9.15232 q^{57} +(-1.40347 - 2.43088i) q^{58} +(-2.09451 + 3.62779i) q^{59} +(0.966847 - 1.67463i) q^{60} +(5.65778 + 9.79957i) q^{61} +8.54301 q^{62} +(4.46335 - 1.01931i) q^{63} +1.00000 q^{64} +(3.25927 + 5.64522i) q^{65} +(-2.05625 + 3.56153i) q^{66} +(6.61841 - 11.4634i) q^{67} +(-2.08850 - 3.61738i) q^{68} +1.12676 q^{69} +(3.08875 + 3.32806i) q^{70} +14.3054 q^{71} +(0.865209 + 1.49859i) q^{72} +(2.13049 - 3.69011i) q^{73} +(-4.46765 + 7.73820i) q^{74} +(-1.15763 - 2.00507i) q^{75} +8.12271 q^{76} +(-6.56903 - 7.07798i) q^{77} +4.27979 q^{78} +(6.72409 + 11.6465i) q^{79} +(-0.858079 + 1.48624i) q^{80} +(0.407198 - 0.705288i) q^{81} +(-5.02592 - 8.70515i) q^{82} +9.43848 q^{83} +(2.90629 - 0.663720i) q^{84} +7.16839 q^{85} +(-4.05534 - 7.02406i) q^{86} +(1.58137 - 2.73901i) q^{87} +(1.82493 - 3.16087i) q^{88} +(-1.97918 - 3.42805i) q^{89} -2.96967 q^{90} +(-2.96094 + 9.60333i) q^{91} -1.00000 q^{92} +(4.81295 + 8.33627i) q^{93} +(1.47568 - 2.55596i) q^{94} +(-6.96993 + 12.0723i) q^{95} +(0.563379 + 0.975800i) q^{96} +0.121105 q^{97} +(-0.521390 + 6.98056i) q^{98} +6.31577 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 3 q^{3} - 4 q^{4} - 7 q^{5} + 6 q^{6} - q^{7} + 8 q^{8} + q^{9} - 7 q^{10} - 2 q^{11} - 3 q^{12} + 2 q^{13} - q^{14} + 18 q^{15} - 4 q^{16} - 5 q^{17} + q^{18} - 11 q^{19} + 14 q^{20}+ \cdots + 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.353553 − 0.612372i
\(3\) 0.563379 − 0.975800i 0.325267 − 0.563379i −0.656300 − 0.754500i \(-0.727879\pi\)
0.981566 + 0.191122i \(0.0612126\pi\)
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −0.858079 − 1.48624i −0.383745 − 0.664665i 0.607849 − 0.794052i \(-0.292032\pi\)
−0.991594 + 0.129387i \(0.958699\pi\)
\(6\) −1.12676 −0.459997
\(7\) 0.779537 − 2.52830i 0.294637 − 0.955609i
\(8\) 1.00000 0.353553
\(9\) 0.865209 + 1.49859i 0.288403 + 0.499529i
\(10\) −0.858079 + 1.48624i −0.271349 + 0.469989i
\(11\) 1.82493 − 3.16087i 0.550236 − 0.953037i −0.448021 − 0.894023i \(-0.647871\pi\)
0.998257 − 0.0590137i \(-0.0187956\pi\)
\(12\) 0.563379 + 0.975800i 0.162633 + 0.281689i
\(13\) −3.79833 −1.05347 −0.526733 − 0.850031i \(-0.676583\pi\)
−0.526733 + 0.850031i \(0.676583\pi\)
\(14\) −2.57934 + 0.589053i −0.689359 + 0.157431i
\(15\) −1.93369 −0.499278
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −2.08850 + 3.61738i −0.506535 + 0.877345i 0.493436 + 0.869782i \(0.335741\pi\)
−0.999971 + 0.00756264i \(0.997593\pi\)
\(18\) 0.865209 − 1.49859i 0.203932 − 0.353220i
\(19\) −4.06135 − 7.03447i −0.931739 − 1.61382i −0.780349 − 0.625345i \(-0.784958\pi\)
−0.151390 − 0.988474i \(-0.548375\pi\)
\(20\) 1.71616 0.383745
\(21\) −2.02795 − 2.18506i −0.442534 − 0.476820i
\(22\) −3.64985 −0.778151
\(23\) 0.500000 + 0.866025i 0.104257 + 0.180579i
\(24\) 0.563379 − 0.975800i 0.114999 − 0.199184i
\(25\) 1.02740 − 1.77951i 0.205480 − 0.355902i
\(26\) 1.89916 + 3.28945i 0.372457 + 0.645114i
\(27\) 5.33003 1.02577
\(28\) 1.79981 + 1.93925i 0.340132 + 0.366484i
\(29\) 2.80694 0.521235 0.260618 − 0.965442i \(-0.416074\pi\)
0.260618 + 0.965442i \(0.416074\pi\)
\(30\) 0.966847 + 1.67463i 0.176521 + 0.305744i
\(31\) −4.27150 + 7.39846i −0.767185 + 1.32880i 0.171899 + 0.985115i \(0.445010\pi\)
−0.939084 + 0.343688i \(0.888324\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) −2.05625 − 3.56153i −0.357947 − 0.619982i
\(34\) 4.17700 0.716349
\(35\) −4.42656 + 1.01091i −0.748226 + 0.170875i
\(36\) −1.73042 −0.288403
\(37\) −4.46765 − 7.73820i −0.734477 − 1.27215i −0.954952 − 0.296759i \(-0.904094\pi\)
0.220475 − 0.975393i \(-0.429239\pi\)
\(38\) −4.06135 + 7.03447i −0.658839 + 1.14114i
\(39\) −2.13990 + 3.70641i −0.342658 + 0.593501i
\(40\) −0.858079 − 1.48624i −0.135674 − 0.234995i
\(41\) 10.0518 1.56983 0.784917 − 0.619601i \(-0.212706\pi\)
0.784917 + 0.619601i \(0.212706\pi\)
\(42\) −0.878349 + 2.84878i −0.135532 + 0.439577i
\(43\) 8.11069 1.23687 0.618434 − 0.785837i \(-0.287767\pi\)
0.618434 + 0.785837i \(0.287767\pi\)
\(44\) 1.82493 + 3.16087i 0.275118 + 0.476518i
\(45\) 1.48484 − 2.57181i 0.221346 − 0.383383i
\(46\) 0.500000 − 0.866025i 0.0737210 − 0.127688i
\(47\) 1.47568 + 2.55596i 0.215250 + 0.372825i 0.953350 − 0.301867i \(-0.0976099\pi\)
−0.738100 + 0.674692i \(0.764277\pi\)
\(48\) −1.12676 −0.162633
\(49\) −5.78464 − 3.94181i −0.826378 − 0.563116i
\(50\) −2.05480 −0.290593
\(51\) 2.35323 + 4.07591i 0.329518 + 0.570742i
\(52\) 1.89916 − 3.28945i 0.263367 − 0.456165i
\(53\) −1.52740 + 2.64553i −0.209804 + 0.363392i −0.951653 − 0.307176i \(-0.900616\pi\)
0.741848 + 0.670568i \(0.233949\pi\)
\(54\) −2.66502 − 4.61594i −0.362663 − 0.628150i
\(55\) −6.26373 −0.844601
\(56\) 0.779537 − 2.52830i 0.104170 − 0.337859i
\(57\) −9.15232 −1.21225
\(58\) −1.40347 − 2.43088i −0.184284 − 0.319190i
\(59\) −2.09451 + 3.62779i −0.272682 + 0.472299i −0.969548 − 0.244903i \(-0.921244\pi\)
0.696866 + 0.717201i \(0.254577\pi\)
\(60\) 0.966847 − 1.67463i 0.124819 − 0.216194i
\(61\) 5.65778 + 9.79957i 0.724405 + 1.25471i 0.959218 + 0.282666i \(0.0912188\pi\)
−0.234813 + 0.972040i \(0.575448\pi\)
\(62\) 8.54301 1.08496
\(63\) 4.46335 − 1.01931i 0.562329 − 0.128421i
\(64\) 1.00000 0.125000
\(65\) 3.25927 + 5.64522i 0.404262 + 0.700203i
\(66\) −2.05625 + 3.56153i −0.253107 + 0.438394i
\(67\) 6.61841 − 11.4634i 0.808567 − 1.40048i −0.105290 − 0.994442i \(-0.533577\pi\)
0.913856 − 0.406037i \(-0.133090\pi\)
\(68\) −2.08850 − 3.61738i −0.253268 − 0.438672i
\(69\) 1.12676 0.135646
\(70\) 3.08875 + 3.32806i 0.369177 + 0.397780i
\(71\) 14.3054 1.69773 0.848867 − 0.528607i \(-0.177285\pi\)
0.848867 + 0.528607i \(0.177285\pi\)
\(72\) 0.865209 + 1.49859i 0.101966 + 0.176610i
\(73\) 2.13049 − 3.69011i 0.249355 − 0.431895i −0.713992 − 0.700154i \(-0.753115\pi\)
0.963347 + 0.268259i \(0.0864482\pi\)
\(74\) −4.46765 + 7.73820i −0.519354 + 0.899547i
\(75\) −1.15763 − 2.00507i −0.133672 − 0.231526i
\(76\) 8.12271 0.931739
\(77\) −6.56903 − 7.07798i −0.748610 − 0.806611i
\(78\) 4.27979 0.484591
\(79\) 6.72409 + 11.6465i 0.756519 + 1.31033i 0.944615 + 0.328179i \(0.106435\pi\)
−0.188096 + 0.982151i \(0.560232\pi\)
\(80\) −0.858079 + 1.48624i −0.0959362 + 0.166166i
\(81\) 0.407198 − 0.705288i 0.0452442 − 0.0783653i
\(82\) −5.02592 − 8.70515i −0.555020 − 0.961323i
\(83\) 9.43848 1.03601 0.518004 − 0.855378i \(-0.326675\pi\)
0.518004 + 0.855378i \(0.326675\pi\)
\(84\) 2.90629 − 0.663720i 0.317103 − 0.0724177i
\(85\) 7.16839 0.777521
\(86\) −4.05534 − 7.02406i −0.437299 − 0.757424i
\(87\) 1.58137 − 2.73901i 0.169540 − 0.293653i
\(88\) 1.82493 − 3.16087i 0.194538 − 0.336949i
\(89\) −1.97918 − 3.42805i −0.209793 − 0.363372i 0.741856 − 0.670559i \(-0.233946\pi\)
−0.951649 + 0.307187i \(0.900612\pi\)
\(90\) −2.96967 −0.313031
\(91\) −2.96094 + 9.60333i −0.310391 + 1.00670i
\(92\) −1.00000 −0.104257
\(93\) 4.81295 + 8.33627i 0.499079 + 0.864431i
\(94\) 1.47568 − 2.55596i 0.152205 − 0.263627i
\(95\) −6.96993 + 12.0723i −0.715100 + 1.23859i
\(96\) 0.563379 + 0.975800i 0.0574996 + 0.0995922i
\(97\) 0.121105 0.0122964 0.00614819 − 0.999981i \(-0.498043\pi\)
0.00614819 + 0.999981i \(0.498043\pi\)
\(98\) −0.521390 + 6.98056i −0.0526683 + 0.705143i
\(99\) 6.31577 0.634759
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 322.2.e.a.277.4 yes 8
7.2 even 3 inner 322.2.e.a.93.4 ✓ 8
7.3 odd 6 2254.2.a.x.1.4 4
7.4 even 3 2254.2.a.z.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.e.a.93.4 ✓ 8 7.2 even 3 inner
322.2.e.a.277.4 yes 8 1.1 even 1 trivial
2254.2.a.x.1.4 4 7.3 odd 6
2254.2.a.z.1.1 4 7.4 even 3