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Properties

Label 322.2.e.a.93.1
Level $322$
Weight $2$
Character 322.93
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 93.1
Root \(-1.03075 - 1.78531i\) of defining polynomial
Character \(\chi\) \(=\) 322.93
Dual form 322.2.e.a.277.1

$q$-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+(-0.500000 + 0.866025i) q^{2} +(-1.28821 - 2.23124i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(-0.663319 + 1.14890i) q^{5} +2.57641 q^{6} +(-1.46157 + 2.20541i) q^{7} +1.00000 q^{8} +(-1.81896 + 3.15053i) q^{9} +(-0.663319 - 1.14890i) q^{10} +(-1.04567 - 1.81115i) q^{11} +(-1.28821 + 2.23124i) q^{12} +3.11142 q^{13} +(-1.17915 - 2.36846i) q^{14} +3.41797 q^{15} +(-0.500000 + 0.866025i) q^{16} +(3.47460 + 6.01818i) q^{17} +(-1.81896 - 3.15053i) q^{18} +(-3.88326 + 6.72601i) q^{19} +1.32664 q^{20} +(6.80360 + 0.420095i) q^{21} +2.09133 q^{22} +(0.500000 - 0.866025i) q^{23} +(-1.28821 - 2.23124i) q^{24} +(1.62002 + 2.80595i) q^{25} +(-1.55571 + 2.69457i) q^{26} +1.64353 q^{27} +(2.64072 + 0.163054i) q^{28} +1.15845 q^{29} +(-1.70898 + 2.96005i) q^{30} +(4.35694 + 7.54644i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(-2.69407 + 4.66626i) q^{33} -6.94919 q^{34} +(-1.56431 - 3.14209i) q^{35} +3.63791 q^{36} +(1.65472 - 2.86606i) q^{37} +(-3.88326 - 6.72601i) q^{38} +(-4.00815 - 6.94232i) q^{39} +(-0.663319 + 1.14890i) q^{40} -4.26425 q^{41} +(-3.76561 + 5.68204i) q^{42} -8.36716 q^{43} +(-1.04567 + 1.81115i) q^{44} +(-2.41310 - 4.17961i) q^{45} +(0.500000 + 0.866025i) q^{46} +(1.74065 - 3.01490i) q^{47} +2.57641 q^{48} +(-2.72763 - 6.44671i) q^{49} -3.24003 q^{50} +(8.95200 - 15.5053i) q^{51} +(-1.55571 - 2.69457i) q^{52} +(-2.12002 - 3.67198i) q^{53} +(-0.821765 + 1.42334i) q^{54} +2.77444 q^{55} +(-1.46157 + 2.20541i) q^{56} +20.0098 q^{57} +(-0.579223 + 1.00324i) q^{58} +(-4.59225 - 7.95401i) q^{59} +(-1.70898 - 2.96005i) q^{60} +(-6.01934 + 10.4258i) q^{61} -8.71388 q^{62} +(-4.28965 - 8.61625i) q^{63} +1.00000 q^{64} +(-2.06386 + 3.57471i) q^{65} +(-2.69407 - 4.66626i) q^{66} +(3.63160 + 6.29011i) q^{67} +(3.47460 - 6.01818i) q^{68} -2.57641 q^{69} +(3.50328 + 0.216314i) q^{70} -7.41708 q^{71} +(-1.81896 + 3.15053i) q^{72} +(2.18402 + 3.78284i) q^{73} +(1.65472 + 2.86606i) q^{74} +(4.17383 - 7.22929i) q^{75} +7.76653 q^{76} +(5.52262 + 0.341000i) q^{77} +8.01631 q^{78} +(0.398625 - 0.690439i) q^{79} +(-0.663319 - 1.14890i) q^{80} +(3.33966 + 5.78446i) q^{81} +(2.13212 - 3.69295i) q^{82} +10.3746 q^{83} +(-3.03798 - 6.10213i) q^{84} -9.21906 q^{85} +(4.18358 - 7.24617i) q^{86} +(-1.49232 - 2.58477i) q^{87} +(-1.04567 - 1.81115i) q^{88} +(-8.32132 + 14.4129i) q^{89} +4.82619 q^{90} +(-4.54756 + 6.86194i) q^{91} -1.00000 q^{92} +(11.2253 - 19.4428i) q^{93} +(1.74065 + 3.01490i) q^{94} +(-5.15168 - 8.92298i) q^{95} +(-1.28821 + 2.23124i) q^{96} +6.65800 q^{97} +(6.94683 + 0.861161i) q^{98} +7.60808 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 \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

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

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
\(3\) −1.28821 − 2.23124i −0.743747 − 1.28821i −0.950778 − 0.309873i \(-0.899713\pi\)
0.207031 − 0.978334i \(-0.433620\pi\)
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) −0.663319 + 1.14890i −0.296645 + 0.513804i −0.975366 − 0.220592i \(-0.929201\pi\)
0.678721 + 0.734396i \(0.262535\pi\)
\(6\) 2.57641 1.05182
\(7\) −1.46157 + 2.20541i −0.552422 + 0.833565i
\(8\) 1.00000 0.353553
\(9\) −1.81896 + 3.15053i −0.606319 + 1.05018i
\(10\) −0.663319 − 1.14890i −0.209760 − 0.363315i
\(11\) −1.04567 − 1.81115i −0.315280 − 0.546081i 0.664217 − 0.747540i \(-0.268765\pi\)
−0.979497 + 0.201459i \(0.935432\pi\)
\(12\) −1.28821 + 2.23124i −0.371873 + 0.644104i
\(13\) 3.11142 0.862952 0.431476 − 0.902124i \(-0.357993\pi\)
0.431476 + 0.902124i \(0.357993\pi\)
\(14\) −1.17915 − 2.36846i −0.315142 − 0.632997i
\(15\) 3.41797 0.882516
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 3.47460 + 6.01818i 0.842713 + 1.45962i 0.887593 + 0.460629i \(0.152376\pi\)
−0.0448794 + 0.998992i \(0.514290\pi\)
\(18\) −1.81896 − 3.15053i −0.428732 − 0.742586i
\(19\) −3.88326 + 6.72601i −0.890882 + 1.54305i −0.0520619 + 0.998644i \(0.516579\pi\)
−0.838820 + 0.544409i \(0.816754\pi\)
\(20\) 1.32664 0.296645
\(21\) 6.80360 + 0.420095i 1.48467 + 0.0916722i
\(22\) 2.09133 0.445873
\(23\) 0.500000 − 0.866025i 0.104257 − 0.180579i
\(24\) −1.28821 − 2.23124i −0.262954 − 0.455450i
\(25\) 1.62002 + 2.80595i 0.324003 + 0.561190i
\(26\) −1.55571 + 2.69457i −0.305100 + 0.528448i
\(27\) 1.64353 0.316298
\(28\) 2.64072 + 0.163054i 0.499050 + 0.0308143i
\(29\) 1.15845 0.215118 0.107559 − 0.994199i \(-0.465697\pi\)
0.107559 + 0.994199i \(0.465697\pi\)
\(30\) −1.70898 + 2.96005i −0.312016 + 0.540428i
\(31\) 4.35694 + 7.54644i 0.782530 + 1.35538i 0.930464 + 0.366384i \(0.119404\pi\)
−0.147934 + 0.988997i \(0.547262\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) −2.69407 + 4.66626i −0.468977 + 0.812292i
\(34\) −6.94919 −1.19178
\(35\) −1.56431 − 3.14209i −0.264416 − 0.531110i
\(36\) 3.63791 0.606319
\(37\) 1.65472 − 2.86606i 0.272034 − 0.471177i −0.697348 − 0.716732i \(-0.745637\pi\)
0.969383 + 0.245555i \(0.0789702\pi\)
\(38\) −3.88326 − 6.72601i −0.629949 − 1.09110i
\(39\) −4.00815 − 6.94232i −0.641818 − 1.11166i
\(40\) −0.663319 + 1.14890i −0.104880 + 0.181657i
\(41\) −4.26425 −0.665964 −0.332982 − 0.942933i \(-0.608055\pi\)
−0.332982 + 0.942933i \(0.608055\pi\)
\(42\) −3.76561 + 5.68204i −0.581046 + 0.876758i
\(43\) −8.36716 −1.27598 −0.637990 − 0.770045i \(-0.720234\pi\)
−0.637990 + 0.770045i \(0.720234\pi\)
\(44\) −1.04567 + 1.81115i −0.157640 + 0.273040i
\(45\) −2.41310 − 4.17961i −0.359723 − 0.623059i
\(46\) 0.500000 + 0.866025i 0.0737210 + 0.127688i
\(47\) 1.74065 − 3.01490i 0.253900 − 0.439768i −0.710696 − 0.703499i \(-0.751620\pi\)
0.964596 + 0.263731i \(0.0849532\pi\)
\(48\) 2.57641 0.371873
\(49\) −2.72763 − 6.44671i −0.389661 − 0.920958i
\(50\) −3.24003 −0.458210
\(51\) 8.95200 − 15.5053i 1.25353 − 2.17118i
\(52\) −1.55571 − 2.69457i −0.215738 − 0.373669i
\(53\) −2.12002 − 3.67198i −0.291207 − 0.504385i 0.682889 − 0.730522i \(-0.260723\pi\)
−0.974095 + 0.226138i \(0.927390\pi\)
\(54\) −0.821765 + 1.42334i −0.111828 + 0.193692i
\(55\) 2.77444 0.374105
\(56\) −1.46157 + 2.20541i −0.195311 + 0.294710i
\(57\) 20.0098 2.65036
\(58\) −0.579223 + 1.00324i −0.0760557 + 0.131732i
\(59\) −4.59225 − 7.95401i −0.597860 − 1.03552i −0.993136 − 0.116961i \(-0.962685\pi\)
0.395277 − 0.918562i \(-0.370649\pi\)
\(60\) −1.70898 − 2.96005i −0.220629 − 0.382140i
\(61\) −6.01934 + 10.4258i −0.770698 + 1.33489i 0.166483 + 0.986044i \(0.446759\pi\)
−0.937181 + 0.348844i \(0.886574\pi\)
\(62\) −8.71388 −1.10666
\(63\) −4.28965 − 8.61625i −0.540446 − 1.08555i
\(64\) 1.00000 0.125000
\(65\) −2.06386 + 3.57471i −0.255991 + 0.443389i
\(66\) −2.69407 − 4.66626i −0.331617 − 0.574377i
\(67\) 3.63160 + 6.29011i 0.443670 + 0.768459i 0.997959 − 0.0638656i \(-0.0203429\pi\)
−0.554288 + 0.832325i \(0.687010\pi\)
\(68\) 3.47460 − 6.01818i 0.421357 − 0.729811i
\(69\) −2.57641 −0.310164
\(70\) 3.50328 + 0.216314i 0.418722 + 0.0258544i
\(71\) −7.41708 −0.880245 −0.440123 − 0.897938i \(-0.645065\pi\)
−0.440123 + 0.897938i \(0.645065\pi\)
\(72\) −1.81896 + 3.15053i −0.214366 + 0.371293i
\(73\) 2.18402 + 3.78284i 0.255621 + 0.442748i 0.965064 − 0.262015i \(-0.0843868\pi\)
−0.709443 + 0.704763i \(0.751054\pi\)
\(74\) 1.65472 + 2.86606i 0.192357 + 0.333173i
\(75\) 4.17383 − 7.22929i 0.481953 − 0.834767i
\(76\) 7.76653 0.890882
\(77\) 5.52262 + 0.341000i 0.629361 + 0.0388606i
\(78\) 8.01631 0.907668
\(79\) 0.398625 − 0.690439i 0.0448488 − 0.0776805i −0.842730 − 0.538337i \(-0.819053\pi\)
0.887578 + 0.460657i \(0.152386\pi\)
\(80\) −0.663319 − 1.14890i −0.0741613 − 0.128451i
\(81\) 3.33966 + 5.78446i 0.371074 + 0.642718i
\(82\) 2.13212 − 3.69295i 0.235454 − 0.407818i
\(83\) 10.3746 1.13876 0.569381 − 0.822074i \(-0.307183\pi\)
0.569381 + 0.822074i \(0.307183\pi\)
\(84\) −3.03798 − 6.10213i −0.331471 − 0.665797i
\(85\) −9.21906 −0.999947
\(86\) 4.18358 − 7.24617i 0.451127 − 0.781375i
\(87\) −1.49232 − 2.58477i −0.159993 − 0.277117i
\(88\) −1.04567 − 1.81115i −0.111468 − 0.193069i
\(89\) −8.32132 + 14.4129i −0.882058 + 1.52777i −0.0330095 + 0.999455i \(0.510509\pi\)
−0.849049 + 0.528315i \(0.822824\pi\)
\(90\) 4.82619 0.508725
\(91\) −4.54756 + 6.86194i −0.476713 + 0.719327i
\(92\) −1.00000 −0.104257
\(93\) 11.2253 − 19.4428i 1.16401 − 2.01612i
\(94\) 1.74065 + 3.01490i 0.179534 + 0.310963i
\(95\) −5.15168 − 8.92298i −0.528552 − 0.915478i
\(96\) −1.28821 + 2.23124i −0.131477 + 0.227725i
\(97\) 6.65800 0.676018 0.338009 − 0.941143i \(-0.390247\pi\)
0.338009 + 0.941143i \(0.390247\pi\)
\(98\) 6.94683 + 0.861161i 0.701735 + 0.0869904i
\(99\) 7.60808 0.764641
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.93.1 ✓ 8
7.2 even 3 2254.2.a.z.1.4 4
7.4 even 3 inner 322.2.e.a.277.1 yes 8
7.5 odd 6 2254.2.a.x.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.e.a.93.1 ✓ 8 1.1 even 1 trivial
322.2.e.a.277.1 yes 8 7.4 even 3 inner
2254.2.a.x.1.1 4 7.5 odd 6
2254.2.a.z.1.4 4 7.2 even 3