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Properties

Label 363.2.e.j.148.1
Level $363$
Weight $2$
Character 363.148
Analytic conductor $2.899$
Analytic rank $0$
Dimension $4$
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] = [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 = [4,3,-1,3] 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: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 148.1
Root \(-0.309017 + 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 363.148
Dual form 363.2.e.j.130.1

$q$-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+(0.190983 + 0.587785i) q^{2} +(-0.809017 - 0.587785i) q^{3} +(1.30902 - 0.951057i) q^{4} +(-0.809017 + 2.48990i) q^{5} +(0.190983 - 0.587785i) q^{6} +(2.42705 - 1.76336i) q^{7} +(1.80902 + 1.31433i) q^{8} +(0.309017 + 0.951057i) q^{9} -1.61803 q^{10} -1.61803 q^{12} +(-0.545085 - 1.67760i) q^{13} +(1.50000 + 1.08981i) q^{14} +(2.11803 - 1.53884i) q^{15} +(0.572949 - 1.76336i) q^{16} +(-0.500000 + 1.53884i) q^{17} +(-0.500000 + 0.363271i) q^{18} +(4.73607 + 3.44095i) q^{19} +(1.30902 + 4.02874i) q^{20} -3.00000 q^{21} +3.47214 q^{23} +(-0.690983 - 2.12663i) q^{24} +(-1.50000 - 1.08981i) q^{25} +(0.881966 - 0.640786i) q^{26} +(0.309017 - 0.951057i) q^{27} +(1.50000 - 4.61653i) q^{28} +(3.61803 - 2.62866i) q^{29} +(1.30902 + 0.951057i) q^{30} +(0.881966 + 2.71441i) q^{31} +5.61803 q^{32} -1.00000 q^{34} +(2.42705 + 7.46969i) q^{35} +(1.30902 + 0.951057i) q^{36} +(-0.190983 + 0.138757i) q^{37} +(-1.11803 + 3.44095i) q^{38} +(-0.545085 + 1.67760i) q^{39} +(-4.73607 + 3.44095i) q^{40} +(-9.66312 - 7.02067i) q^{41} +(-0.572949 - 1.76336i) q^{42} -6.23607 q^{43} -2.61803 q^{45} +(0.663119 + 2.04087i) q^{46} +(-1.30902 - 0.951057i) q^{47} +(-1.50000 + 1.08981i) q^{48} +(0.618034 - 1.90211i) q^{49} +(0.354102 - 1.08981i) q^{50} +(1.30902 - 0.951057i) q^{51} +(-2.30902 - 1.67760i) q^{52} +(-2.97214 - 9.14729i) q^{53} +0.618034 q^{54} +6.70820 q^{56} +(-1.80902 - 5.56758i) q^{57} +(2.23607 + 1.62460i) q^{58} +(-8.35410 + 6.06961i) q^{59} +(1.30902 - 4.02874i) q^{60} +(-2.42705 + 7.46969i) q^{61} +(-1.42705 + 1.03681i) q^{62} +(2.42705 + 1.76336i) q^{63} +(-0.0729490 - 0.224514i) q^{64} +4.61803 q^{65} -9.56231 q^{67} +(0.809017 + 2.48990i) q^{68} +(-2.80902 - 2.04087i) q^{69} +(-3.92705 + 2.85317i) q^{70} +(-1.71885 + 5.29007i) q^{71} +(-0.690983 + 2.12663i) q^{72} +(-2.61803 + 1.90211i) q^{73} +(-0.118034 - 0.0857567i) q^{74} +(0.572949 + 1.76336i) q^{75} +9.47214 q^{76} -1.09017 q^{78} +(-2.92705 - 9.00854i) q^{79} +(3.92705 + 2.85317i) q^{80} +(-0.809017 + 0.587785i) q^{81} +(2.28115 - 7.02067i) q^{82} +(-0.218847 + 0.673542i) q^{83} +(-3.92705 + 2.85317i) q^{84} +(-3.42705 - 2.48990i) q^{85} +(-1.19098 - 3.66547i) q^{86} -4.47214 q^{87} +0.527864 q^{89} +(-0.500000 - 1.53884i) q^{90} +(-4.28115 - 3.11044i) q^{91} +(4.54508 - 3.30220i) q^{92} +(0.881966 - 2.71441i) q^{93} +(0.309017 - 0.951057i) q^{94} +(-12.3992 + 9.00854i) q^{95} +(-4.54508 - 3.30220i) q^{96} +(-4.33688 - 13.3475i) q^{97} +1.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} + 3 q^{6} + 3 q^{7} + 5 q^{8} - q^{9} - 2 q^{10} - 2 q^{12} + 9 q^{13} + 6 q^{14} + 4 q^{15} + 9 q^{16} - 2 q^{17} - 2 q^{18} + 10 q^{19} + 3 q^{20} - 12 q^{21}+ \cdots - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.190983 + 0.587785i 0.135045 + 0.415627i 0.995597 − 0.0937362i \(-0.0298810\pi\)
−0.860552 + 0.509363i \(0.829881\pi\)
\(3\) −0.809017 − 0.587785i −0.467086 − 0.339358i
\(4\) 1.30902 − 0.951057i 0.654508 − 0.475528i
\(5\) −0.809017 + 2.48990i −0.361803 + 1.11352i 0.590155 + 0.807290i \(0.299067\pi\)
−0.951959 + 0.306227i \(0.900933\pi\)
\(6\) 0.190983 − 0.587785i 0.0779685 − 0.239962i
\(7\) 2.42705 − 1.76336i 0.917339 − 0.666486i −0.0255212 − 0.999674i \(-0.508125\pi\)
0.942860 + 0.333188i \(0.108125\pi\)
\(8\) 1.80902 + 1.31433i 0.639584 + 0.464685i
\(9\) 0.309017 + 0.951057i 0.103006 + 0.317019i
\(10\) −1.61803 −0.511667
\(11\) 0 0
\(12\) −1.61803 −0.467086
\(13\) −0.545085 − 1.67760i −0.151179 − 0.465282i 0.846574 − 0.532270i \(-0.178661\pi\)
−0.997754 + 0.0669881i \(0.978661\pi\)
\(14\) 1.50000 + 1.08981i 0.400892 + 0.291265i
\(15\) 2.11803 − 1.53884i 0.546874 − 0.397327i
\(16\) 0.572949 − 1.76336i 0.143237 − 0.440839i
\(17\) −0.500000 + 1.53884i −0.121268 + 0.373224i −0.993203 − 0.116398i \(-0.962865\pi\)
0.871935 + 0.489622i \(0.162865\pi\)
\(18\) −0.500000 + 0.363271i −0.117851 + 0.0856239i
\(19\) 4.73607 + 3.44095i 1.08653 + 0.789409i 0.978810 − 0.204772i \(-0.0656454\pi\)
0.107719 + 0.994181i \(0.465645\pi\)
\(20\) 1.30902 + 4.02874i 0.292705 + 0.900854i
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 3.47214 0.723990 0.361995 − 0.932180i \(-0.382096\pi\)
0.361995 + 0.932180i \(0.382096\pi\)
\(24\) −0.690983 − 2.12663i −0.141046 − 0.434096i
\(25\) −1.50000 − 1.08981i −0.300000 − 0.217963i
\(26\) 0.881966 − 0.640786i 0.172968 − 0.125668i
\(27\) 0.309017 − 0.951057i 0.0594703 − 0.183031i
\(28\) 1.50000 − 4.61653i 0.283473 − 0.872441i
\(29\) 3.61803 − 2.62866i 0.671852 − 0.488129i −0.198793 − 0.980042i \(-0.563702\pi\)
0.870645 + 0.491912i \(0.163702\pi\)
\(30\) 1.30902 + 0.951057i 0.238993 + 0.173638i
\(31\) 0.881966 + 2.71441i 0.158406 + 0.487523i 0.998490 − 0.0549331i \(-0.0174946\pi\)
−0.840084 + 0.542456i \(0.817495\pi\)
\(32\) 5.61803 0.993137
\(33\) 0 0
\(34\) −1.00000 −0.171499
\(35\) 2.42705 + 7.46969i 0.410246 + 1.26261i
\(36\) 1.30902 + 0.951057i 0.218169 + 0.158509i
\(37\) −0.190983 + 0.138757i −0.0313974 + 0.0228116i −0.603373 − 0.797459i \(-0.706177\pi\)
0.571976 + 0.820270i \(0.306177\pi\)
\(38\) −1.11803 + 3.44095i −0.181369 + 0.558197i
\(39\) −0.545085 + 1.67760i −0.0872835 + 0.268631i
\(40\) −4.73607 + 3.44095i −0.748838 + 0.544063i
\(41\) −9.66312 − 7.02067i −1.50913 − 1.09644i −0.966563 − 0.256428i \(-0.917454\pi\)
−0.542562 − 0.840015i \(-0.682546\pi\)
\(42\) −0.572949 − 1.76336i −0.0884080 − 0.272092i
\(43\) −6.23607 −0.950991 −0.475496 − 0.879718i \(-0.657731\pi\)
−0.475496 + 0.879718i \(0.657731\pi\)
\(44\) 0 0
\(45\) −2.61803 −0.390273
\(46\) 0.663119 + 2.04087i 0.0977716 + 0.300910i
\(47\) −1.30902 − 0.951057i −0.190940 − 0.138726i 0.488208 − 0.872727i \(-0.337651\pi\)
−0.679148 + 0.734001i \(0.737651\pi\)
\(48\) −1.50000 + 1.08981i −0.216506 + 0.157301i
\(49\) 0.618034 − 1.90211i 0.0882906 − 0.271730i
\(50\) 0.354102 − 1.08981i 0.0500776 − 0.154123i
\(51\) 1.30902 − 0.951057i 0.183299 − 0.133175i
\(52\) −2.30902 − 1.67760i −0.320203 − 0.232641i
\(53\) −2.97214 − 9.14729i −0.408254 − 1.25648i −0.918147 − 0.396240i \(-0.870315\pi\)
0.509893 − 0.860238i \(-0.329685\pi\)
\(54\) 0.618034 0.0841038
\(55\) 0 0
\(56\) 6.70820 0.896421
\(57\) −1.80902 − 5.56758i −0.239610 − 0.737444i
\(58\) 2.23607 + 1.62460i 0.293610 + 0.213320i
\(59\) −8.35410 + 6.06961i −1.08761 + 0.790196i −0.978994 − 0.203888i \(-0.934642\pi\)
−0.108617 + 0.994084i \(0.534642\pi\)
\(60\) 1.30902 − 4.02874i 0.168993 − 0.520108i
\(61\) −2.42705 + 7.46969i −0.310752 + 0.956396i 0.666716 + 0.745312i \(0.267699\pi\)
−0.977468 + 0.211084i \(0.932301\pi\)
\(62\) −1.42705 + 1.03681i −0.181236 + 0.131675i
\(63\) 2.42705 + 1.76336i 0.305780 + 0.222162i
\(64\) −0.0729490 − 0.224514i −0.00911863 − 0.0280642i
\(65\) 4.61803 0.572797
\(66\) 0 0
\(67\) −9.56231 −1.16822 −0.584111 − 0.811674i \(-0.698557\pi\)
−0.584111 + 0.811674i \(0.698557\pi\)
\(68\) 0.809017 + 2.48990i 0.0981077 + 0.301945i
\(69\) −2.80902 − 2.04087i −0.338166 − 0.245692i
\(70\) −3.92705 + 2.85317i −0.469372 + 0.341019i
\(71\) −1.71885 + 5.29007i −0.203990 + 0.627815i 0.795764 + 0.605607i \(0.207070\pi\)
−0.999753 + 0.0222083i \(0.992930\pi\)
\(72\) −0.690983 + 2.12663i −0.0814331 + 0.250625i
\(73\) −2.61803 + 1.90211i −0.306418 + 0.222625i −0.730358 − 0.683065i \(-0.760647\pi\)
0.423940 + 0.905690i \(0.360647\pi\)
\(74\) −0.118034 − 0.0857567i −0.0137212 − 0.00996902i
\(75\) 0.572949 + 1.76336i 0.0661585 + 0.203615i
\(76\) 9.47214 1.08653
\(77\) 0 0
\(78\) −1.09017 −0.123437
\(79\) −2.92705 − 9.00854i −0.329319 − 1.01354i −0.969453 − 0.245276i \(-0.921121\pi\)
0.640134 − 0.768263i \(-0.278879\pi\)
\(80\) 3.92705 + 2.85317i 0.439058 + 0.318994i
\(81\) −0.809017 + 0.587785i −0.0898908 + 0.0653095i
\(82\) 2.28115 − 7.02067i 0.251911 − 0.775303i
\(83\) −0.218847 + 0.673542i −0.0240216 + 0.0739308i −0.962349 − 0.271818i \(-0.912375\pi\)
0.938327 + 0.345749i \(0.112375\pi\)
\(84\) −3.92705 + 2.85317i −0.428476 + 0.311306i
\(85\) −3.42705 − 2.48990i −0.371716 − 0.270067i
\(86\) −1.19098 − 3.66547i −0.128427 − 0.395258i
\(87\) −4.47214 −0.479463
\(88\) 0 0
\(89\) 0.527864 0.0559535 0.0279767 − 0.999609i \(-0.491094\pi\)
0.0279767 + 0.999609i \(0.491094\pi\)
\(90\) −0.500000 − 1.53884i −0.0527046 − 0.162208i
\(91\) −4.28115 − 3.11044i −0.448787 − 0.326063i
\(92\) 4.54508 − 3.30220i 0.473858 − 0.344278i
\(93\) 0.881966 − 2.71441i 0.0914556 − 0.281471i
\(94\) 0.309017 − 0.951057i 0.0318727 − 0.0980940i
\(95\) −12.3992 + 9.00854i −1.27213 + 0.924256i
\(96\) −4.54508 − 3.30220i −0.463881 − 0.337029i
\(97\) −4.33688 − 13.3475i −0.440344 − 1.35524i −0.887510 − 0.460788i \(-0.847567\pi\)
0.447167 − 0.894451i \(-0.352433\pi\)
\(98\) 1.23607 0.124862
\(99\) 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.j.148.1 4
11.2 odd 10 33.2.e.a.31.1 yes 4
11.3 even 5 363.2.a.e.1.2 2
11.4 even 5 363.2.e.c.124.1 4
11.5 even 5 363.2.e.c.202.1 4
11.6 odd 10 363.2.e.h.202.1 4
11.7 odd 10 363.2.e.h.124.1 4
11.8 odd 10 363.2.a.h.1.1 2
11.9 even 5 inner 363.2.e.j.130.1 4
11.10 odd 2 33.2.e.a.16.1 ✓ 4
33.2 even 10 99.2.f.b.64.1 4
33.8 even 10 1089.2.a.m.1.2 2
33.14 odd 10 1089.2.a.s.1.1 2
33.32 even 2 99.2.f.b.82.1 4
44.3 odd 10 5808.2.a.bm.1.1 2
44.19 even 10 5808.2.a.bl.1.1 2
44.35 even 10 528.2.y.f.97.1 4
44.43 even 2 528.2.y.f.49.1 4
55.2 even 20 825.2.bx.b.724.1 8
55.13 even 20 825.2.bx.b.724.2 8
55.14 even 10 9075.2.a.bv.1.1 2
55.19 odd 10 9075.2.a.x.1.2 2
55.24 odd 10 825.2.n.f.526.1 4
55.32 even 4 825.2.bx.b.49.2 8
55.43 even 4 825.2.bx.b.49.1 8
55.54 odd 2 825.2.n.f.676.1 4
99.2 even 30 891.2.n.a.757.1 8
99.13 odd 30 891.2.n.d.460.1 8
99.32 even 6 891.2.n.a.379.1 8
99.43 odd 6 891.2.n.d.676.1 8
99.65 even 6 891.2.n.a.676.1 8
99.68 even 30 891.2.n.a.460.1 8
99.76 odd 6 891.2.n.d.379.1 8
99.79 odd 30 891.2.n.d.757.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.a.16.1 ✓ 4 11.10 odd 2
33.2.e.a.31.1 yes 4 11.2 odd 10
99.2.f.b.64.1 4 33.2 even 10
99.2.f.b.82.1 4 33.32 even 2
363.2.a.e.1.2 2 11.3 even 5
363.2.a.h.1.1 2 11.8 odd 10
363.2.e.c.124.1 4 11.4 even 5
363.2.e.c.202.1 4 11.5 even 5
363.2.e.h.124.1 4 11.7 odd 10
363.2.e.h.202.1 4 11.6 odd 10
363.2.e.j.130.1 4 11.9 even 5 inner
363.2.e.j.148.1 4 1.1 even 1 trivial
528.2.y.f.49.1 4 44.43 even 2
528.2.y.f.97.1 4 44.35 even 10
825.2.n.f.526.1 4 55.24 odd 10
825.2.n.f.676.1 4 55.54 odd 2
825.2.bx.b.49.1 8 55.43 even 4
825.2.bx.b.49.2 8 55.32 even 4
825.2.bx.b.724.1 8 55.2 even 20
825.2.bx.b.724.2 8 55.13 even 20
891.2.n.a.379.1 8 99.32 even 6
891.2.n.a.460.1 8 99.68 even 30
891.2.n.a.676.1 8 99.65 even 6
891.2.n.a.757.1 8 99.2 even 30
891.2.n.d.379.1 8 99.76 odd 6
891.2.n.d.460.1 8 99.13 odd 30
891.2.n.d.676.1 8 99.43 odd 6
891.2.n.d.757.1 8 99.79 odd 30
1089.2.a.m.1.2 2 33.8 even 10
1089.2.a.s.1.1 2 33.14 odd 10
5808.2.a.bl.1.1 2 44.19 even 10
5808.2.a.bm.1.1 2 44.3 odd 10
9075.2.a.x.1.2 2 55.19 odd 10
9075.2.a.bv.1.1 2 55.14 even 10