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Properties

Label 363.2.f.i.233.3
Level $363$
Weight $2$
Character 363.233
Analytic conductor $2.899$
Analytic rank $0$
Dimension $32$
Inner twists $16$

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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(161,363)] 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("363.161"); 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([5, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.f (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,4,-16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 233.3
Character \(\chi\) \(=\) 363.233
Dual form 363.2.f.i.215.3

$q$-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+(-0.465371 + 1.43226i) q^{2} +(0.479216 - 1.66444i) q^{3} +(-0.216775 - 0.157497i) q^{4} +(2.76692 - 0.899027i) q^{5} +(2.16090 + 1.46094i) q^{6} +(1.66251 - 2.28825i) q^{7} +(-2.11025 + 1.53319i) q^{8} +(-2.54070 - 1.59525i) q^{9} +4.38134i q^{10} +(-0.366025 + 0.285334i) q^{12} +(0.852694 + 0.277057i) q^{13} +(2.50369 + 3.44603i) q^{14} +(-0.170423 - 5.03620i) q^{15} +(-1.37948 - 4.24561i) q^{16} +(-0.806046 - 2.48075i) q^{17} +(3.46719 - 2.89658i) q^{18} +(-1.43977 - 1.98168i) q^{19} +(-0.741394 - 0.240894i) q^{20} +(-3.01194 - 3.86370i) q^{21} +(1.54063 + 4.24712i) q^{24} +(2.80252 - 2.03615i) q^{25} +(-0.793638 + 1.09235i) q^{26} +(-3.87274 + 3.46438i) q^{27} +(-0.720782 + 0.234196i) q^{28} +(6.98368 + 5.07394i) q^{29} +(7.29247 + 2.09961i) q^{30} +(-3.15078 + 9.69712i) q^{31} +1.50597 q^{32} +3.92820 q^{34} +(2.54283 - 7.82603i) q^{35} +(0.299516 + 0.745963i) q^{36} +(2.86060 + 2.07835i) q^{37} +(3.50832 - 1.13992i) q^{38} +(0.869768 - 1.28649i) q^{39} +(-4.46053 + 6.13939i) q^{40} +(3.65507 - 2.65556i) q^{41} +(6.93551 - 2.51584i) q^{42} +4.24264i q^{43} +(-8.46410 - 2.12976i) q^{45} +(-1.25184 - 1.72302i) q^{47} +(-7.72763 + 0.261500i) q^{48} +(-0.309017 - 0.951057i) q^{49} +(1.61209 + 4.96151i) q^{50} +(-4.51533 + 0.152797i) q^{51} +(-0.141208 - 0.194356i) q^{52} +(6.27524 + 2.03895i) q^{53} +(-3.15964 - 7.15900i) q^{54} +7.37772i q^{56} +(-3.98834 + 1.44676i) q^{57} +(-10.5172 + 7.64121i) q^{58} +(0.335431 - 0.461681i) q^{59} +(-0.756240 + 1.11856i) q^{60} +(-9.67880 + 3.14483i) q^{61} +(-12.4225 - 9.02551i) q^{62} +(-7.87426 + 3.16164i) q^{63} +(2.05813 - 6.33428i) q^{64} +2.60842 q^{65} +2.00000 q^{67} +(-0.215979 + 0.664716i) q^{68} +(10.0256 + 7.28401i) q^{70} +(-9.04216 + 2.93797i) q^{71} +(7.80735 - 0.529001i) q^{72} +(-1.88524 + 2.59481i) q^{73} +(-4.30798 + 3.12993i) q^{74} +(-2.04603 - 5.64037i) q^{75} +0.656339i q^{76} +(1.43782 + 1.84443i) q^{78} +(-6.98881 - 2.27080i) q^{79} +(-7.63384 - 10.5071i) q^{80} +(3.91036 + 8.10611i) q^{81} +(2.10250 + 6.47084i) q^{82} +(-1.95277 - 6.01000i) q^{83} +(0.0443951 + 1.31193i) q^{84} +(-4.46053 - 6.13939i) q^{85} +(-6.07658 - 1.97440i) q^{86} +(11.7919 - 9.19239i) q^{87} -12.4168i q^{89} +(6.98933 - 11.1317i) q^{90} +(2.05158 - 1.49056i) q^{91} +(14.6303 + 9.89129i) q^{93} +(3.05038 - 0.991130i) q^{94} +(-5.76532 - 4.18875i) q^{95} +(0.721685 - 2.50660i) q^{96} +(1.77130 - 5.45150i) q^{97} +1.50597 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 4 q^{3} - 16 q^{4} + 8 q^{9} + 16 q^{12} - 4 q^{15} + 8 q^{16} + 4 q^{27} + 40 q^{31} - 96 q^{34} + 40 q^{36} + 56 q^{37} - 64 q^{42} - 160 q^{45} - 28 q^{48} + 8 q^{49} - 104 q^{58} + 28 q^{60}+ \cdots - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.465371 + 1.43226i −0.329067 + 1.01276i 0.640505 + 0.767954i \(0.278725\pi\)
−0.969571 + 0.244809i \(0.921275\pi\)
\(3\) 0.479216 − 1.66444i 0.276675 − 0.960963i
\(4\) −0.216775 − 0.157497i −0.108388 − 0.0787483i
\(5\) 2.76692 − 0.899027i 1.23740 − 0.402057i 0.384013 − 0.923328i \(-0.374542\pi\)
0.853392 + 0.521270i \(0.174542\pi\)
\(6\) 2.16090 + 1.46094i 0.882184 + 0.596428i
\(7\) 1.66251 − 2.28825i 0.628369 − 0.864876i −0.369560 − 0.929207i \(-0.620491\pi\)
0.997929 + 0.0643314i \(0.0204915\pi\)
\(8\) −2.11025 + 1.53319i −0.746088 + 0.542065i
\(9\) −2.54070 − 1.59525i −0.846902 − 0.531750i
\(10\) 4.38134i 1.38550i
\(11\) 0 0
\(12\) −0.366025 + 0.285334i −0.105662 + 0.0823689i
\(13\) 0.852694 + 0.277057i 0.236495 + 0.0768418i 0.424867 − 0.905256i \(-0.360321\pi\)
−0.188372 + 0.982098i \(0.560321\pi\)
\(14\) 2.50369 + 3.44603i 0.669139 + 0.920991i
\(15\) −0.170423 − 5.03620i −0.0440030 − 1.30034i
\(16\) −1.37948 − 4.24561i −0.344871 − 1.06140i
\(17\) −0.806046 − 2.48075i −0.195495 − 0.601671i −0.999970 − 0.00768550i \(-0.997554\pi\)
0.804476 − 0.593986i \(-0.202446\pi\)
\(18\) 3.46719 − 2.89658i 0.817224 − 0.682730i
\(19\) −1.43977 − 1.98168i −0.330307 − 0.454628i 0.611272 − 0.791420i \(-0.290658\pi\)
−0.941579 + 0.336792i \(0.890658\pi\)
\(20\) −0.741394 − 0.240894i −0.165781 − 0.0538654i
\(21\) −3.01194 − 3.86370i −0.657260 − 0.843129i
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 1.54063 + 4.24712i 0.314480 + 0.866939i
\(25\) 2.80252 − 2.03615i 0.560503 − 0.407230i
\(26\) −0.793638 + 1.09235i −0.155645 + 0.214227i
\(27\) −3.87274 + 3.46438i −0.745309 + 0.666720i
\(28\) −0.720782 + 0.234196i −0.136215 + 0.0442589i
\(29\) 6.98368 + 5.07394i 1.29684 + 0.942207i 0.999919 − 0.0126960i \(-0.00404139\pi\)
0.296917 + 0.954903i \(0.404041\pi\)
\(30\) 7.29247 + 2.09961i 1.33142 + 0.383334i
\(31\) −3.15078 + 9.69712i −0.565898 + 1.74165i 0.0993719 + 0.995050i \(0.468317\pi\)
−0.665270 + 0.746603i \(0.731683\pi\)
\(32\) 1.50597 0.266221
\(33\) 0 0
\(34\) 3.92820 0.673681
\(35\) 2.54283 − 7.82603i 0.429817 − 1.32284i
\(36\) 0.299516 + 0.745963i 0.0499194 + 0.124327i
\(37\) 2.86060 + 2.07835i 0.470280 + 0.341678i 0.797550 − 0.603252i \(-0.206129\pi\)
−0.327270 + 0.944931i \(0.606129\pi\)
\(38\) 3.50832 − 1.13992i 0.569124 − 0.184920i
\(39\) 0.869768 − 1.28649i 0.139274 − 0.206003i
\(40\) −4.46053 + 6.13939i −0.705272 + 0.970723i
\(41\) 3.65507 − 2.65556i 0.570826 − 0.414729i −0.264579 − 0.964364i \(-0.585233\pi\)
0.835405 + 0.549635i \(0.185233\pi\)
\(42\) 6.93551 − 2.51584i 1.07017 − 0.388203i
\(43\) 4.24264i 0.646997i 0.946229 + 0.323498i \(0.104859\pi\)
−0.946229 + 0.323498i \(0.895141\pi\)
\(44\) 0 0
\(45\) −8.46410 − 2.12976i −1.26175 − 0.317487i
\(46\) 0 0
\(47\) −1.25184 − 1.72302i −0.182600 − 0.251328i 0.707898 − 0.706315i \(-0.249644\pi\)
−0.890498 + 0.454987i \(0.849644\pi\)
\(48\) −7.72763 + 0.261500i −1.11539 + 0.0377443i
\(49\) −0.309017 − 0.951057i −0.0441453 − 0.135865i
\(50\) 1.61209 + 4.96151i 0.227984 + 0.701663i
\(51\) −4.51533 + 0.152797i −0.632273 + 0.0213959i
\(52\) −0.141208 − 0.194356i −0.0195820 − 0.0269523i
\(53\) 6.27524 + 2.03895i 0.861970 + 0.280071i 0.706451 − 0.707762i \(-0.250295\pi\)
0.155519 + 0.987833i \(0.450295\pi\)
\(54\) −3.15964 − 7.15900i −0.429973 − 0.974217i
\(55\) 0 0
\(56\) 7.37772i 0.985890i
\(57\) −3.98834 + 1.44676i −0.528269 + 0.191628i
\(58\) −10.5172 + 7.64121i −1.38098 + 1.00334i
\(59\) 0.335431 − 0.461681i 0.0436694 − 0.0601057i −0.786624 − 0.617432i \(-0.788173\pi\)
0.830293 + 0.557327i \(0.188173\pi\)
\(60\) −0.756240 + 1.11856i −0.0976302 + 0.144406i
\(61\) −9.67880 + 3.14483i −1.23924 + 0.402655i −0.854055 − 0.520183i \(-0.825864\pi\)
−0.385189 + 0.922838i \(0.625864\pi\)
\(62\) −12.4225 − 9.02551i −1.57766 − 1.14624i
\(63\) −7.87426 + 3.16164i −0.992064 + 0.398330i
\(64\) 2.05813 − 6.33428i 0.257266 − 0.791785i
\(65\) 2.60842 0.323535
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −0.215979 + 0.664716i −0.0261913 + 0.0806086i
\(69\) 0 0
\(70\) 10.0256 + 7.28401i 1.19829 + 0.870606i
\(71\) −9.04216 + 2.93797i −1.07311 + 0.348673i −0.791697 − 0.610914i \(-0.790802\pi\)
−0.281410 + 0.959588i \(0.590802\pi\)
\(72\) 7.80735 − 0.529001i 0.920106 − 0.0623434i
\(73\) −1.88524 + 2.59481i −0.220651 + 0.303700i −0.904964 − 0.425489i \(-0.860102\pi\)
0.684313 + 0.729188i \(0.260102\pi\)
\(74\) −4.30798 + 3.12993i −0.500793 + 0.363847i
\(75\) −2.04603 − 5.64037i −0.236255 − 0.651294i
\(76\) 0.656339i 0.0752872i
\(77\) 0 0
\(78\) 1.43782 + 1.84443i 0.162801 + 0.208841i
\(79\) −6.98881 − 2.27080i −0.786303 − 0.255485i −0.111774 − 0.993734i \(-0.535653\pi\)
−0.674529 + 0.738249i \(0.735653\pi\)
\(80\) −7.63384 − 10.5071i −0.853490 − 1.17473i
\(81\) 3.91036 + 8.10611i 0.434485 + 0.900679i
\(82\) 2.10250 + 6.47084i 0.232183 + 0.714585i
\(83\) −1.95277 − 6.01000i −0.214344 − 0.659683i −0.999200 − 0.0400041i \(-0.987263\pi\)
0.784856 − 0.619679i \(-0.212737\pi\)
\(84\) 0.0443951 + 1.31193i 0.00484390 + 0.143143i
\(85\) −4.46053 − 6.13939i −0.483812 − 0.665911i
\(86\) −6.07658 − 1.97440i −0.655255 − 0.212905i
\(87\) 11.7919 − 9.19239i 1.26423 − 0.985527i
\(88\) 0 0
\(89\) − 12.4168i − 1.31618i −0.752940 − 0.658089i \(-0.771365\pi\)
0.752940 − 0.658089i \(-0.228635\pi\)
\(90\) 6.98933 − 11.1317i 0.736740 − 1.17338i
\(91\) 2.05158 − 1.49056i 0.215065 − 0.156254i
\(92\) 0 0
\(93\) 14.6303 + 9.89129i 1.51710 + 1.02568i
\(94\) 3.05038 − 0.991130i 0.314623 − 0.102227i
\(95\) −5.76532 − 4.18875i −0.591510 − 0.429757i
\(96\) 0.721685 − 2.50660i 0.0736567 − 0.255828i
\(97\) 1.77130 − 5.45150i 0.179848 − 0.553516i −0.819973 − 0.572402i \(-0.806012\pi\)
0.999822 + 0.0188854i \(0.00601177\pi\)
\(98\) 1.50597 0.152126
\(99\) 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.f.i.233.3 32
3.2 odd 2 inner 363.2.f.i.233.6 32
11.2 odd 10 inner 363.2.f.i.239.3 32
11.3 even 5 inner 363.2.f.i.161.6 32
11.4 even 5 363.2.d.e.362.4 yes 8
11.5 even 5 inner 363.2.f.i.215.4 32
11.6 odd 10 inner 363.2.f.i.215.6 32
11.7 odd 10 363.2.d.e.362.6 yes 8
11.8 odd 10 inner 363.2.f.i.161.4 32
11.9 even 5 inner 363.2.f.i.239.5 32
11.10 odd 2 inner 363.2.f.i.233.5 32
33.2 even 10 inner 363.2.f.i.239.6 32
33.5 odd 10 inner 363.2.f.i.215.5 32
33.8 even 10 inner 363.2.f.i.161.5 32
33.14 odd 10 inner 363.2.f.i.161.3 32
33.17 even 10 inner 363.2.f.i.215.3 32
33.20 odd 10 inner 363.2.f.i.239.4 32
33.26 odd 10 363.2.d.e.362.5 yes 8
33.29 even 10 363.2.d.e.362.3 ✓ 8
33.32 even 2 inner 363.2.f.i.233.4 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.2.d.e.362.3 ✓ 8 33.29 even 10
363.2.d.e.362.4 yes 8 11.4 even 5
363.2.d.e.362.5 yes 8 33.26 odd 10
363.2.d.e.362.6 yes 8 11.7 odd 10
363.2.f.i.161.3 32 33.14 odd 10 inner
363.2.f.i.161.4 32 11.8 odd 10 inner
363.2.f.i.161.5 32 33.8 even 10 inner
363.2.f.i.161.6 32 11.3 even 5 inner
363.2.f.i.215.3 32 33.17 even 10 inner
363.2.f.i.215.4 32 11.5 even 5 inner
363.2.f.i.215.5 32 33.5 odd 10 inner
363.2.f.i.215.6 32 11.6 odd 10 inner
363.2.f.i.233.3 32 1.1 even 1 trivial
363.2.f.i.233.4 32 33.32 even 2 inner
363.2.f.i.233.5 32 11.10 odd 2 inner
363.2.f.i.233.6 32 3.2 odd 2 inner
363.2.f.i.239.3 32 11.2 odd 10 inner
363.2.f.i.239.4 32 33.20 odd 10 inner
363.2.f.i.239.5 32 11.9 even 5 inner
363.2.f.i.239.6 32 33.2 even 10 inner