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Properties

Label 322.2.e.a.277.2
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 277.2
Root \(0.346911 - 0.600868i\) of defining polynomial
Character \(\chi\) \(=\) 322.277
Dual form 322.2.e.a.93.2

$q$-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} +(-0.873734 + 1.51335i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-2.13304 - 3.69453i) q^{5} +1.74747 q^{6} +(1.89234 + 1.84906i) q^{7} +1.00000 q^{8} +(-0.0268230 - 0.0464588i) q^{9} +(-2.13304 + 3.69453i) q^{10} +(-1.59438 + 2.76155i) q^{11} +(-0.873734 - 1.51335i) q^{12} -4.60948 q^{13} +(0.655163 - 2.56335i) q^{14} +7.45484 q^{15} +(-0.500000 - 0.866025i) q^{16} +(-1.57939 + 2.73559i) q^{17} +(-0.0268230 + 0.0464588i) q^{18} +(2.26815 + 3.92856i) q^{19} +4.26608 q^{20} +(-4.45169 + 1.24819i) q^{21} +3.18876 q^{22} +(0.500000 + 0.866025i) q^{23} +(-0.873734 + 1.51335i) q^{24} +(-6.59971 + 11.4310i) q^{25} +(2.30474 + 3.99192i) q^{26} -5.14866 q^{27} +(-2.54751 + 0.714287i) q^{28} -3.70737 q^{29} +(-3.72742 - 6.45608i) q^{30} +(-1.61805 + 2.80255i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(-2.78613 - 4.82572i) q^{33} +3.15879 q^{34} +(2.79498 - 10.9355i) q^{35} +0.0536460 q^{36} +(3.62328 + 6.27570i) q^{37} +(2.26815 - 3.92856i) q^{38} +(4.02746 - 6.97576i) q^{39} +(-2.13304 - 3.69453i) q^{40} +5.11454 q^{41} +(3.30681 + 3.23118i) q^{42} -2.29605 q^{43} +(-1.59438 - 2.76155i) q^{44} +(-0.114429 + 0.198197i) q^{45} +(0.500000 - 0.866025i) q^{46} +(-2.84899 - 4.93459i) q^{47} +1.74747 q^{48} +(0.161936 + 6.99813i) q^{49} +13.1994 q^{50} +(-2.75994 - 4.78036i) q^{51} +(2.30474 - 3.99192i) q^{52} +(6.09971 - 10.5650i) q^{53} +(2.57433 + 4.45887i) q^{54} +13.6035 q^{55} +(1.89234 + 1.84906i) q^{56} -7.92705 q^{57} +(1.85368 + 3.21068i) q^{58} +(-0.459266 + 0.795473i) q^{59} +(-3.72742 + 6.45608i) q^{60} +(-6.74448 - 11.6818i) q^{61} +3.23611 q^{62} +(0.0351469 - 0.137513i) q^{63} +1.00000 q^{64} +(9.83220 + 17.0299i) q^{65} +(-2.78613 + 4.82572i) q^{66} +(-2.37788 + 4.11861i) q^{67} +(-1.57939 - 2.73559i) q^{68} -1.74747 q^{69} +(-10.8679 + 3.04721i) q^{70} +3.61960 q^{71} +(-0.0268230 - 0.0464588i) q^{72} +(6.68525 - 11.5792i) q^{73} +(3.62328 - 6.27570i) q^{74} +(-11.5328 - 19.9754i) q^{75} -4.53631 q^{76} +(-8.12339 + 2.27769i) q^{77} -8.05492 q^{78} +(3.71036 + 6.42653i) q^{79} +(-2.13304 + 3.69453i) q^{80} +(4.57903 - 7.93111i) q^{81} +(-2.55727 - 4.42932i) q^{82} -9.36524 q^{83} +(1.14488 - 4.47937i) q^{84} +13.4756 q^{85} +(1.14803 + 1.98844i) q^{86} +(3.23926 - 5.61056i) q^{87} +(-1.59438 + 2.76155i) q^{88} +(2.61155 + 4.52334i) q^{89} +0.228858 q^{90} +(-8.72272 - 8.52321i) q^{91} -1.00000 q^{92} +(-2.82750 - 4.89737i) q^{93} +(-2.84899 + 4.93459i) q^{94} +(9.67612 - 16.7595i) q^{95} +(-0.873734 - 1.51335i) q^{96} -5.74459 q^{97} +(5.97959 - 3.63930i) q^{98} +0.171064 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
\(3\) −0.873734 + 1.51335i −0.504451 + 0.873734i 0.495536 + 0.868587i \(0.334972\pi\)
−0.999987 + 0.00514686i \(0.998362\pi\)
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −2.13304 − 3.69453i −0.953924 − 1.65225i −0.736811 − 0.676099i \(-0.763669\pi\)
−0.217113 − 0.976146i \(-0.569664\pi\)
\(6\) 1.74747 0.713401
\(7\) 1.89234 + 1.84906i 0.715239 + 0.698880i
\(8\) 1.00000 0.353553
\(9\) −0.0268230 − 0.0464588i −0.00894100 − 0.0154863i
\(10\) −2.13304 + 3.69453i −0.674526 + 1.16831i
\(11\) −1.59438 + 2.76155i −0.480724 + 0.832638i −0.999755 − 0.0221173i \(-0.992959\pi\)
0.519032 + 0.854755i \(0.326293\pi\)
\(12\) −0.873734 − 1.51335i −0.252225 − 0.436867i
\(13\) −4.60948 −1.27844 −0.639219 − 0.769024i \(-0.720742\pi\)
−0.639219 + 0.769024i \(0.720742\pi\)
\(14\) 0.655163 − 2.56335i 0.175100 − 0.685084i
\(15\) 7.45484 1.92483
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −1.57939 + 2.73559i −0.383059 + 0.663478i −0.991498 − 0.130123i \(-0.958463\pi\)
0.608439 + 0.793601i \(0.291796\pi\)
\(18\) −0.0268230 + 0.0464588i −0.00632224 + 0.0109504i
\(19\) 2.26815 + 3.92856i 0.520350 + 0.901273i 0.999720 + 0.0236597i \(0.00753182\pi\)
−0.479370 + 0.877613i \(0.659135\pi\)
\(20\) 4.26608 0.953924
\(21\) −4.45169 + 1.24819i −0.971438 + 0.272378i
\(22\) 3.18876 0.679846
\(23\) 0.500000 + 0.866025i 0.104257 + 0.180579i
\(24\) −0.873734 + 1.51335i −0.178350 + 0.308912i
\(25\) −6.59971 + 11.4310i −1.31994 + 2.28621i
\(26\) 2.30474 + 3.99192i 0.451996 + 0.782881i
\(27\) −5.14866 −0.990860
\(28\) −2.54751 + 0.714287i −0.481434 + 0.134988i
\(29\) −3.70737 −0.688441 −0.344221 − 0.938889i \(-0.611857\pi\)
−0.344221 + 0.938889i \(0.611857\pi\)
\(30\) −3.72742 − 6.45608i −0.680531 − 1.17871i
\(31\) −1.61805 + 2.80255i −0.290611 + 0.503353i −0.973954 − 0.226744i \(-0.927192\pi\)
0.683343 + 0.730097i \(0.260525\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) −2.78613 − 4.82572i −0.485003 − 0.840049i
\(34\) 3.15879 0.541727
\(35\) 2.79498 − 10.9355i 0.472437 − 1.84843i
\(36\) 0.0536460 0.00894100
\(37\) 3.62328 + 6.27570i 0.595663 + 1.03172i 0.993453 + 0.114242i \(0.0364439\pi\)
−0.397790 + 0.917477i \(0.630223\pi\)
\(38\) 2.26815 − 3.92856i 0.367943 − 0.637296i
\(39\) 4.02746 − 6.97576i 0.644909 − 1.11702i
\(40\) −2.13304 − 3.69453i −0.337263 − 0.584157i
\(41\) 5.11454 0.798757 0.399378 − 0.916786i \(-0.369226\pi\)
0.399378 + 0.916786i \(0.369226\pi\)
\(42\) 3.30681 + 3.23118i 0.510252 + 0.498582i
\(43\) −2.29605 −0.350145 −0.175072 − 0.984556i \(-0.556016\pi\)
−0.175072 + 0.984556i \(0.556016\pi\)
\(44\) −1.59438 − 2.76155i −0.240362 − 0.416319i
\(45\) −0.114429 + 0.198197i −0.0170581 + 0.0295454i
\(46\) 0.500000 − 0.866025i 0.0737210 − 0.127688i
\(47\) −2.84899 − 4.93459i −0.415567 − 0.719783i 0.579921 − 0.814673i \(-0.303084\pi\)
−0.995488 + 0.0948895i \(0.969750\pi\)
\(48\) 1.74747 0.252225
\(49\) 0.161936 + 6.99813i 0.0231338 + 0.999732i
\(50\) 13.1994 1.86668
\(51\) −2.75994 − 4.78036i −0.386469 − 0.669384i
\(52\) 2.30474 − 3.99192i 0.319610 − 0.553580i
\(53\) 6.09971 − 10.5650i 0.837860 − 1.45122i −0.0538200 − 0.998551i \(-0.517140\pi\)
0.891680 − 0.452666i \(-0.149527\pi\)
\(54\) 2.57433 + 4.45887i 0.350322 + 0.606775i
\(55\) 13.6035 1.83430
\(56\) 1.89234 + 1.84906i 0.252875 + 0.247091i
\(57\) −7.92705 −1.04996
\(58\) 1.85368 + 3.21068i 0.243401 + 0.421582i
\(59\) −0.459266 + 0.795473i −0.0597914 + 0.103562i −0.894372 − 0.447325i \(-0.852377\pi\)
0.834580 + 0.550886i \(0.185710\pi\)
\(60\) −3.72742 + 6.45608i −0.481208 + 0.833476i
\(61\) −6.74448 − 11.6818i −0.863542 − 1.49570i −0.868488 − 0.495711i \(-0.834908\pi\)
0.00494545 − 0.999988i \(-0.498426\pi\)
\(62\) 3.23611 0.410986
\(63\) 0.0351469 − 0.137513i 0.00442809 − 0.0173251i
\(64\) 1.00000 0.125000
\(65\) 9.83220 + 17.0299i 1.21953 + 2.11229i
\(66\) −2.78613 + 4.82572i −0.342949 + 0.594005i
\(67\) −2.37788 + 4.11861i −0.290505 + 0.503169i −0.973929 − 0.226852i \(-0.927156\pi\)
0.683425 + 0.730021i \(0.260490\pi\)
\(68\) −1.57939 − 2.73559i −0.191530 − 0.331739i
\(69\) −1.74747 −0.210370
\(70\) −10.8679 + 3.04721i −1.29896 + 0.364211i
\(71\) 3.61960 0.429568 0.214784 − 0.976662i \(-0.431095\pi\)
0.214784 + 0.976662i \(0.431095\pi\)
\(72\) −0.0268230 − 0.0464588i −0.00316112 − 0.00547522i
\(73\) 6.68525 − 11.5792i 0.782449 − 1.35524i −0.148062 − 0.988978i \(-0.547304\pi\)
0.930511 − 0.366263i \(-0.119363\pi\)
\(74\) 3.62328 − 6.27570i 0.421197 − 0.729535i
\(75\) −11.5328 − 19.9754i −1.33169 − 2.30656i
\(76\) −4.53631 −0.520350
\(77\) −8.12339 + 2.27769i −0.925746 + 0.259567i
\(78\) −8.05492 −0.912040
\(79\) 3.71036 + 6.42653i 0.417448 + 0.723041i 0.995682 − 0.0928298i \(-0.0295913\pi\)
−0.578234 + 0.815871i \(0.696258\pi\)
\(80\) −2.13304 + 3.69453i −0.238481 + 0.413061i
\(81\) 4.57903 − 7.93111i 0.508781 − 0.881235i
\(82\) −2.55727 − 4.42932i −0.282403 − 0.489137i
\(83\) −9.36524 −1.02797 −0.513984 − 0.857800i \(-0.671831\pi\)
−0.513984 + 0.857800i \(0.671831\pi\)
\(84\) 1.14488 − 4.47937i 0.124916 − 0.488740i
\(85\) 13.4756 1.46164
\(86\) 1.14803 + 1.98844i 0.123795 + 0.214419i
\(87\) 3.23926 − 5.61056i 0.347285 − 0.601515i
\(88\) −1.59438 + 2.76155i −0.169961 + 0.294382i
\(89\) 2.61155 + 4.52334i 0.276824 + 0.479473i 0.970594 − 0.240724i \(-0.0773848\pi\)
−0.693770 + 0.720197i \(0.744051\pi\)
\(90\) 0.228858 0.0241238
\(91\) −8.72272 − 8.52321i −0.914389 − 0.893475i
\(92\) −1.00000 −0.104257
\(93\) −2.82750 − 4.89737i −0.293198 − 0.507833i
\(94\) −2.84899 + 4.93459i −0.293850 + 0.508964i
\(95\) 9.67612 − 16.7595i 0.992749 − 1.71949i
\(96\) −0.873734 − 1.51335i −0.0891751 − 0.154456i
\(97\) −5.74459 −0.583275 −0.291637 − 0.956529i \(-0.594200\pi\)
−0.291637 + 0.956529i \(0.594200\pi\)
\(98\) 5.97959 − 3.63930i 0.604030 − 0.367625i
\(99\) 0.171064 0.0171926
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.277.2 yes 8
7.2 even 3 inner 322.2.e.a.93.2 ✓ 8
7.3 odd 6 2254.2.a.x.1.2 4
7.4 even 3 2254.2.a.z.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.e.a.93.2 ✓ 8 7.2 even 3 inner
322.2.e.a.277.2 yes 8 1.1 even 1 trivial
2254.2.a.x.1.2 4 7.3 odd 6
2254.2.a.z.1.3 4 7.4 even 3