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Properties

Label 363.2.e.n.130.3
Level $363$
Weight $2$
Character 363.130
Analytic conductor $2.899$
Analytic rank $0$
Dimension $16$
Inner twists $8$

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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] = [363,2,Mod(124,363)] 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("363.124"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.e (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,-4,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: 16.0.22502537891856000000000000.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 7x^{14} + 45x^{12} + 287x^{10} + 1829x^{8} + 1148x^{6} + 720x^{4} + 448x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 130.3
Root \(0.244830 + 0.753510i\) of defining polynomial
Character \(\chi\) \(=\) 363.130
Dual form 363.2.e.n.148.3

$q$-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+(0.244830 - 0.753510i) q^{2} +(-0.809017 + 0.587785i) q^{3} +(1.11020 + 0.806607i) q^{4} +(1.04209 + 3.20723i) q^{5} +(0.244830 + 0.753510i) q^{6} +(-2.04223 - 1.48377i) q^{7} +(2.16154 - 1.57045i) q^{8} +(0.309017 - 0.951057i) q^{9} +2.67181 q^{10} -1.37228 q^{12} +(-1.80496 + 5.55509i) q^{13} +(-1.61803 + 1.17557i) q^{14} +(-2.72823 - 1.98218i) q^{15} +(0.193976 + 0.596996i) q^{16} +(0.825636 + 2.54105i) q^{17} +(-0.640974 - 0.465695i) q^{18} +(0.760285 - 0.552379i) q^{19} +(-1.43004 + 4.40122i) q^{20} +2.52434 q^{21} +2.00000 q^{23} +(-0.825636 + 2.54105i) q^{24} +(-5.15528 + 3.74553i) q^{25} +(3.74390 + 2.72010i) q^{26} +(0.309017 + 0.951057i) q^{27} +(-1.07047 - 3.29456i) q^{28} +(0.640974 + 0.465695i) q^{29} +(-2.16154 + 1.57045i) q^{30} +(0.502993 - 1.54805i) q^{31} +5.84096 q^{32} +2.11684 q^{34} +(2.63059 - 8.09613i) q^{35} +(1.11020 - 0.806607i) q^{36} +(-4.04508 - 2.93893i) q^{37} +(-0.230083 - 0.708121i) q^{38} +(-1.80496 - 5.55509i) q^{39} +(7.28933 + 5.29601i) q^{40} +(8.80990 - 6.40077i) q^{41} +(0.618034 - 1.90211i) q^{42} -6.63325 q^{43} +3.37228 q^{45} +(0.489660 - 1.50702i) q^{46} +(10.3106 - 7.49107i) q^{47} +(-0.507835 - 0.368964i) q^{48} +(-0.193976 - 0.596996i) q^{49} +(1.56013 + 4.80158i) q^{50} +(-2.16154 - 1.57045i) q^{51} +(-6.48463 + 4.71136i) q^{52} +(-1.27217 + 3.91535i) q^{53} +0.792287 q^{54} -6.74456 q^{56} +(-0.290403 + 0.893769i) q^{57} +(0.507835 - 0.368964i) q^{58} +(4.85410 + 3.52671i) q^{59} +(-1.43004 - 4.40122i) q^{60} +(1.85053 + 5.69534i) q^{61} +(-1.04332 - 0.758020i) q^{62} +(-2.04223 + 1.48377i) q^{63} +(1.04209 - 3.20723i) q^{64} -19.6974 q^{65} -1.11684 q^{67} +(-1.13301 + 3.48703i) q^{68} +(-1.61803 + 1.17557i) q^{69} +(-5.45647 - 3.96435i) q^{70} +(-3.32025 - 10.2187i) q^{71} +(-0.825636 - 2.54105i) q^{72} +(-7.40864 - 5.38269i) q^{73} +(-3.20487 + 2.32847i) q^{74} +(1.96914 - 6.06040i) q^{75} +1.28962 q^{76} -4.62772 q^{78} +(1.26972 - 3.90781i) q^{79} +(-1.71256 + 1.24425i) q^{80} +(-0.809017 - 0.587785i) q^{81} +(-2.66611 - 8.20545i) q^{82} +(-0.580806 - 1.78754i) q^{83} +(2.80252 + 2.03615i) q^{84} +(-7.28933 + 5.29601i) q^{85} +(-1.62402 + 4.99822i) q^{86} -0.792287 q^{87} -0.627719 q^{89} +(0.825636 - 2.54105i) q^{90} +(11.9286 - 8.66664i) q^{91} +(2.22040 + 1.61321i) q^{92} +(0.502993 + 1.54805i) q^{93} +(-3.12025 - 9.60315i) q^{94} +(2.56389 + 1.86278i) q^{95} +(-4.72544 + 3.43323i) q^{96} +(3.24132 - 9.97575i) q^{97} -0.497333 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} - 6 q^{4} - 2 q^{5} - 4 q^{9} + 24 q^{12} - 8 q^{14} - 2 q^{15} - 14 q^{16} + 30 q^{20} + 32 q^{23} - 14 q^{25} + 30 q^{26} - 4 q^{27} - 18 q^{31} - 104 q^{34} - 6 q^{36} - 20 q^{37} - 20 q^{38}+ \cdots + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.244830 − 0.753510i 0.173121 − 0.532812i −0.826422 − 0.563052i \(-0.809627\pi\)
0.999543 + 0.0302400i \(0.00962715\pi\)
\(3\) −0.809017 + 0.587785i −0.467086 + 0.339358i
\(4\) 1.11020 + 0.806607i 0.555099 + 0.403303i
\(5\) 1.04209 + 3.20723i 0.466038 + 1.43432i 0.857673 + 0.514196i \(0.171910\pi\)
−0.391635 + 0.920121i \(0.628090\pi\)
\(6\) 0.244830 + 0.753510i 0.0999515 + 0.307619i
\(7\) −2.04223 − 1.48377i −0.771891 − 0.560812i 0.130643 − 0.991429i \(-0.458296\pi\)
−0.902534 + 0.430618i \(0.858296\pi\)
\(8\) 2.16154 − 1.57045i 0.764221 − 0.555239i
\(9\) 0.309017 − 0.951057i 0.103006 − 0.317019i
\(10\) 2.67181 0.844902
\(11\) 0 0
\(12\) −1.37228 −0.396143
\(13\) −1.80496 + 5.55509i −0.500605 + 1.54070i 0.307431 + 0.951570i \(0.400531\pi\)
−0.808036 + 0.589133i \(0.799469\pi\)
\(14\) −1.61803 + 1.17557i −0.432438 + 0.314184i
\(15\) −2.72823 − 1.98218i −0.704427 − 0.511796i
\(16\) 0.193976 + 0.596996i 0.0484939 + 0.149249i
\(17\) 0.825636 + 2.54105i 0.200246 + 0.616294i 0.999875 + 0.0158000i \(0.00502951\pi\)
−0.799629 + 0.600494i \(0.794970\pi\)
\(18\) −0.640974 − 0.465695i −0.151079 − 0.109765i
\(19\) 0.760285 − 0.552379i 0.174421 − 0.126725i −0.497149 − 0.867665i \(-0.665620\pi\)
0.671571 + 0.740940i \(0.265620\pi\)
\(20\) −1.43004 + 4.40122i −0.319767 + 0.984143i
\(21\) 2.52434 0.550856
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) −0.825636 + 2.54105i −0.168532 + 0.518689i
\(25\) −5.15528 + 3.74553i −1.03106 + 0.749107i
\(26\) 3.74390 + 2.72010i 0.734240 + 0.533456i
\(27\) 0.309017 + 0.951057i 0.0594703 + 0.183031i
\(28\) −1.07047 − 3.29456i −0.202299 − 0.622613i
\(29\) 0.640974 + 0.465695i 0.119026 + 0.0864773i 0.645706 − 0.763586i \(-0.276563\pi\)
−0.526680 + 0.850064i \(0.676563\pi\)
\(30\) −2.16154 + 1.57045i −0.394642 + 0.286724i
\(31\) 0.502993 − 1.54805i 0.0903402 − 0.278038i −0.895671 − 0.444717i \(-0.853304\pi\)
0.986011 + 0.166679i \(0.0533042\pi\)
\(32\) 5.84096 1.03255
\(33\) 0 0
\(34\) 2.11684 0.363036
\(35\) 2.63059 − 8.09613i 0.444651 − 1.36850i
\(36\) 1.11020 − 0.806607i 0.185033 − 0.134434i
\(37\) −4.04508 − 2.93893i −0.665008 − 0.483157i 0.203343 − 0.979108i \(-0.434819\pi\)
−0.868350 + 0.495951i \(0.834819\pi\)
\(38\) −0.230083 − 0.708121i −0.0373243 − 0.114872i
\(39\) −1.80496 − 5.55509i −0.289024 − 0.889526i
\(40\) 7.28933 + 5.29601i 1.15254 + 0.837373i
\(41\) 8.80990 − 6.40077i 1.37588 − 0.999632i 0.378623 − 0.925551i \(-0.376398\pi\)
0.997252 − 0.0740813i \(-0.0236024\pi\)
\(42\) 0.618034 − 1.90211i 0.0953647 − 0.293502i
\(43\) −6.63325 −1.01156 −0.505781 − 0.862662i \(-0.668795\pi\)
−0.505781 + 0.862662i \(0.668795\pi\)
\(44\) 0 0
\(45\) 3.37228 0.502710
\(46\) 0.489660 − 1.50702i 0.0721965 − 0.222198i
\(47\) 10.3106 − 7.49107i 1.50395 − 1.09268i 0.535177 − 0.844740i \(-0.320245\pi\)
0.968774 − 0.247944i \(-0.0797549\pi\)
\(48\) −0.507835 − 0.368964i −0.0732997 − 0.0532553i
\(49\) −0.193976 − 0.596996i −0.0277108 − 0.0852851i
\(50\) 1.56013 + 4.80158i 0.220635 + 0.679045i
\(51\) −2.16154 − 1.57045i −0.302677 − 0.219907i
\(52\) −6.48463 + 4.71136i −0.899256 + 0.653348i
\(53\) −1.27217 + 3.91535i −0.174747 + 0.537815i −0.999622 − 0.0275001i \(-0.991245\pi\)
0.824875 + 0.565315i \(0.191245\pi\)
\(54\) 0.792287 0.107817
\(55\) 0 0
\(56\) −6.74456 −0.901280
\(57\) −0.290403 + 0.893769i −0.0384648 + 0.118383i
\(58\) 0.507835 − 0.368964i 0.0666820 − 0.0484473i
\(59\) 4.85410 + 3.52671i 0.631950 + 0.459139i 0.857075 − 0.515191i \(-0.172279\pi\)
−0.225125 + 0.974330i \(0.572279\pi\)
\(60\) −1.43004 − 4.40122i −0.184618 − 0.568195i
\(61\) 1.85053 + 5.69534i 0.236936 + 0.729214i 0.996859 + 0.0792006i \(0.0252368\pi\)
−0.759923 + 0.650013i \(0.774763\pi\)
\(62\) −1.04332 − 0.758020i −0.132502 − 0.0962686i
\(63\) −2.04223 + 1.48377i −0.257297 + 0.186937i
\(64\) 1.04209 − 3.20723i 0.130262 − 0.400904i
\(65\) −19.6974 −2.44316
\(66\) 0 0
\(67\) −1.11684 −0.136444 −0.0682221 − 0.997670i \(-0.521733\pi\)
−0.0682221 + 0.997670i \(0.521733\pi\)
\(68\) −1.13301 + 3.48703i −0.137397 + 0.422865i
\(69\) −1.61803 + 1.17557i −0.194788 + 0.141522i
\(70\) −5.45647 − 3.96435i −0.652172 − 0.473831i
\(71\) −3.32025 − 10.2187i −0.394041 − 1.21273i −0.929706 − 0.368304i \(-0.879939\pi\)
0.535664 − 0.844431i \(-0.320061\pi\)
\(72\) −0.825636 − 2.54105i −0.0973022 − 0.299465i
\(73\) −7.40864 − 5.38269i −0.867116 − 0.629997i 0.0626954 − 0.998033i \(-0.480030\pi\)
−0.929812 + 0.368036i \(0.880030\pi\)
\(74\) −3.20487 + 2.32847i −0.372558 + 0.270679i
\(75\) 1.96914 − 6.06040i 0.227377 − 0.699795i
\(76\) 1.28962 0.147930
\(77\) 0 0
\(78\) −4.62772 −0.523986
\(79\) 1.26972 − 3.90781i 0.142855 − 0.439663i −0.853874 − 0.520480i \(-0.825753\pi\)
0.996729 + 0.0808173i \(0.0257530\pi\)
\(80\) −1.71256 + 1.24425i −0.191470 + 0.139111i
\(81\) −0.809017 − 0.587785i −0.0898908 − 0.0653095i
\(82\) −2.66611 − 8.20545i −0.294423 − 0.906140i
\(83\) −0.580806 − 1.78754i −0.0637517 − 0.196208i 0.914107 − 0.405472i \(-0.132893\pi\)
−0.977859 + 0.209265i \(0.932893\pi\)
\(84\) 2.80252 + 2.03615i 0.305780 + 0.222162i
\(85\) −7.28933 + 5.29601i −0.790639 + 0.574433i
\(86\) −1.62402 + 4.99822i −0.175123 + 0.538972i
\(87\) −0.792287 −0.0849421
\(88\) 0 0
\(89\) −0.627719 −0.0665380 −0.0332690 − 0.999446i \(-0.510592\pi\)
−0.0332690 + 0.999446i \(0.510592\pi\)
\(90\) 0.825636 − 2.54105i 0.0870297 − 0.267850i
\(91\) 11.9286 − 8.66664i 1.25046 − 0.908510i
\(92\) 2.22040 + 1.61321i 0.231492 + 0.168189i
\(93\) 0.502993 + 1.54805i 0.0521579 + 0.160526i
\(94\) −3.12025 − 9.60315i −0.321830 − 0.990489i
\(95\) 2.56389 + 1.86278i 0.263050 + 0.191117i
\(96\) −4.72544 + 3.43323i −0.482288 + 0.350403i
\(97\) 3.24132 − 9.97575i 0.329106 − 1.01288i −0.640447 − 0.768002i \(-0.721251\pi\)
0.969553 − 0.244882i \(-0.0787491\pi\)
\(98\) −0.497333 −0.0502383
\(99\) 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 363.2.e.n.130.3 16
11.2 odd 10 inner 363.2.e.n.124.3 16
11.3 even 5 inner 363.2.e.n.202.2 16
11.4 even 5 363.2.a.j.1.3 yes 4
11.5 even 5 inner 363.2.e.n.148.3 16
11.6 odd 10 inner 363.2.e.n.148.2 16
11.7 odd 10 363.2.a.j.1.2 ✓ 4
11.8 odd 10 inner 363.2.e.n.202.3 16
11.9 even 5 inner 363.2.e.n.124.2 16
11.10 odd 2 inner 363.2.e.n.130.2 16
33.26 odd 10 1089.2.a.u.1.2 4
33.29 even 10 1089.2.a.u.1.3 4
44.7 even 10 5808.2.a.ck.1.4 4
44.15 odd 10 5808.2.a.ck.1.3 4
55.4 even 10 9075.2.a.cv.1.2 4
55.29 odd 10 9075.2.a.cv.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.2.a.j.1.2 ✓ 4 11.7 odd 10
363.2.a.j.1.3 yes 4 11.4 even 5
363.2.e.n.124.2 16 11.9 even 5 inner
363.2.e.n.124.3 16 11.2 odd 10 inner
363.2.e.n.130.2 16 11.10 odd 2 inner
363.2.e.n.130.3 16 1.1 even 1 trivial
363.2.e.n.148.2 16 11.6 odd 10 inner
363.2.e.n.148.3 16 11.5 even 5 inner
363.2.e.n.202.2 16 11.3 even 5 inner
363.2.e.n.202.3 16 11.8 odd 10 inner
1089.2.a.u.1.2 4 33.26 odd 10
1089.2.a.u.1.3 4 33.29 even 10
5808.2.a.ck.1.3 4 44.15 odd 10
5808.2.a.ck.1.4 4 44.7 even 10
9075.2.a.cv.1.2 4 55.4 even 10
9075.2.a.cv.1.3 4 55.29 odd 10