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Properties

Label 465.2.t.d.119.4
Level $465$
Weight $2$
Character 465.119
Analytic conductor $3.713$
Analytic rank $0$
Dimension $104$
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] = [465,2,Mod(119,465)] 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("465.119"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [104,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(52\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 119.4
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.4

$q$-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-2.25277 q^{2} +(-1.27444 - 1.17295i) q^{3} +3.07497 q^{4} +(-0.0924590 + 2.23416i) q^{5} +(2.87101 + 2.64239i) q^{6} +(-3.62956 - 2.09552i) q^{7} -2.42166 q^{8} +(0.248372 + 2.98970i) q^{9} +(0.208289 - 5.03304i) q^{10} +(-1.60948 - 2.78770i) q^{11} +(-3.91885 - 3.60679i) q^{12} +(1.90775 + 3.30433i) q^{13} +(8.17655 + 4.72073i) q^{14} +(2.73839 - 2.73884i) q^{15} -0.694506 q^{16} +(0.967553 + 0.558617i) q^{17} +(-0.559525 - 6.73511i) q^{18} +(-1.50611 + 2.60865i) q^{19} +(-0.284309 + 6.86996i) q^{20} +(2.16769 + 6.92790i) q^{21} +(3.62579 + 6.28004i) q^{22} +1.62078i q^{23} +(3.08624 + 2.84048i) q^{24} +(-4.98290 - 0.413136i) q^{25} +(-4.29773 - 7.44389i) q^{26} +(3.19024 - 4.10151i) q^{27} +(-11.1608 - 6.44367i) q^{28} +1.01929 q^{29} +(-6.16896 + 6.16997i) q^{30} +(3.32985 - 4.46230i) q^{31} +6.40787 q^{32} +(-1.21866 + 5.44058i) q^{33} +(-2.17967 - 1.25843i) q^{34} +(5.01731 - 7.91524i) q^{35} +(0.763736 + 9.19324i) q^{36} +(0.345854 - 0.599038i) q^{37} +(3.39291 - 5.87669i) q^{38} +(1.44450 - 6.44886i) q^{39} +(0.223904 - 5.41036i) q^{40} +(10.6022 - 6.12120i) q^{41} +(-4.88330 - 15.6070i) q^{42} +(1.99045 - 3.44756i) q^{43} +(-4.94910 - 8.57209i) q^{44} +(-6.70242 + 0.278477i) q^{45} -3.65123i q^{46} +5.63789 q^{47} +(0.885103 + 0.814621i) q^{48} +(5.28245 + 9.14947i) q^{49} +(11.2253 + 0.930699i) q^{50} +(-0.577853 - 1.84681i) q^{51} +(5.86629 + 10.1607i) q^{52} +(3.70980 - 2.14185i) q^{53} +(-7.18687 + 9.23975i) q^{54} +(6.37697 - 3.33808i) q^{55} +(8.78953 + 5.07464i) q^{56} +(4.97926 - 1.55797i) q^{57} -2.29622 q^{58} +(-6.43120 - 3.71306i) q^{59} +(8.42046 - 8.42184i) q^{60} -12.0124i q^{61} +(-7.50138 + 10.0525i) q^{62} +(5.36351 - 11.3718i) q^{63} -13.0464 q^{64} +(-7.55877 + 3.95671i) q^{65} +(2.74535 - 12.2564i) q^{66} +(12.2696 - 7.08384i) q^{67} +(2.97519 + 1.71773i) q^{68} +(1.90109 - 2.06557i) q^{69} +(-11.3028 + 17.8312i) q^{70} +(-9.38271 + 5.41711i) q^{71} +(-0.601471 - 7.24003i) q^{72} +(1.40116 + 2.42688i) q^{73} +(-0.779130 + 1.34949i) q^{74} +(5.86580 + 6.37122i) q^{75} +(-4.63123 + 8.02152i) q^{76} +13.4908i q^{77} +(-3.25413 + 14.5278i) q^{78} +(13.8006 + 7.96777i) q^{79} +(0.0642133 - 1.55163i) q^{80} +(-8.87662 + 1.48512i) q^{81} +(-23.8844 + 13.7897i) q^{82} +(-2.00628 + 1.15833i) q^{83} +(6.66557 + 21.3031i) q^{84} +(-1.33750 + 2.11001i) q^{85} +(-4.48402 + 7.76655i) q^{86} +(-1.29902 - 1.19558i) q^{87} +(3.89760 + 6.75085i) q^{88} +16.0855 q^{89} +(15.0990 - 0.627344i) q^{90} -15.9910i q^{91} +4.98383i q^{92} +(-9.47773 + 1.78117i) q^{93} -12.7009 q^{94} +(-5.68888 - 3.60607i) q^{95} +(-8.16642 - 7.51612i) q^{96} -13.7208i q^{97} +(-11.9001 - 20.6116i) q^{98} +(7.93464 - 5.50425i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 104 q + 56 q^{4} - 12 q^{6} - 2 q^{9} - 20 q^{10} - 24 q^{16} + 12 q^{19} - 6 q^{21} - 48 q^{24} + 8 q^{25} - 40 q^{31} + 108 q^{34} + 36 q^{36} + 52 q^{39} - 76 q^{40} - 4 q^{45} + 80 q^{49} - 20 q^{51}+ \cdots + 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.25277 −1.59295 −0.796474 − 0.604673i \(-0.793304\pi\)
−0.796474 + 0.604673i \(0.793304\pi\)
\(3\) −1.27444 − 1.17295i −0.735796 − 0.677204i
\(4\) 3.07497 1.53748
\(5\) −0.0924590 + 2.23416i −0.0413489 + 0.999145i
\(6\) 2.87101 + 2.64239i 1.17208 + 1.07875i
\(7\) −3.62956 − 2.09552i −1.37184 − 0.792034i −0.380683 − 0.924706i \(-0.624311\pi\)
−0.991160 + 0.132672i \(0.957644\pi\)
\(8\) −2.42166 −0.856185
\(9\) 0.248372 + 2.98970i 0.0827906 + 0.996567i
\(10\) 0.208289 − 5.03304i 0.0658667 − 1.59159i
\(11\) −1.60948 − 2.78770i −0.485276 − 0.840523i 0.514581 − 0.857442i \(-0.327948\pi\)
−0.999857 + 0.0169189i \(0.994614\pi\)
\(12\) −3.91885 − 3.60679i −1.13127 − 1.04119i
\(13\) 1.90775 + 3.30433i 0.529116 + 0.916456i 0.999423 + 0.0339531i \(0.0108097\pi\)
−0.470308 + 0.882503i \(0.655857\pi\)
\(14\) 8.17655 + 4.72073i 2.18527 + 1.26167i
\(15\) 2.73839 − 2.73884i 0.707049 − 0.707165i
\(16\) −0.694506 −0.173626
\(17\) 0.967553 + 0.558617i 0.234666 + 0.135484i 0.612723 − 0.790298i \(-0.290074\pi\)
−0.378057 + 0.925782i \(0.623408\pi\)
\(18\) −0.559525 − 6.73511i −0.131881 − 1.58748i
\(19\) −1.50611 + 2.60865i −0.345524 + 0.598466i −0.985449 − 0.169972i \(-0.945632\pi\)
0.639924 + 0.768438i \(0.278966\pi\)
\(20\) −0.284309 + 6.86996i −0.0635733 + 1.53617i
\(21\) 2.16769 + 6.92790i 0.473028 + 1.51179i
\(22\) 3.62579 + 6.28004i 0.773020 + 1.33891i
\(23\) 1.62078i 0.337955i 0.985620 + 0.168978i \(0.0540465\pi\)
−0.985620 + 0.168978i \(0.945953\pi\)
\(24\) 3.08624 + 2.84048i 0.629977 + 0.579811i
\(25\) −4.98290 − 0.413136i −0.996581 − 0.0826271i
\(26\) −4.29773 − 7.44389i −0.842854 − 1.45987i
\(27\) 3.19024 − 4.10151i 0.613962 − 0.789336i
\(28\) −11.1608 − 6.44367i −2.10919 − 1.21774i
\(29\) 1.01929 0.189277 0.0946387 − 0.995512i \(-0.469830\pi\)
0.0946387 + 0.995512i \(0.469830\pi\)
\(30\) −6.16896 + 6.16997i −1.12629 + 1.12648i
\(31\) 3.32985 − 4.46230i 0.598058 − 0.801453i
\(32\) 6.40787 1.13276
\(33\) −1.21866 + 5.44058i −0.212141 + 0.947084i
\(34\) −2.17967 − 1.25843i −0.373811 − 0.215820i
\(35\) 5.01731 − 7.91524i 0.848081 − 1.33792i
\(36\) 0.763736 + 9.19324i 0.127289 + 1.53221i
\(37\) 0.345854 − 0.599038i 0.0568581 − 0.0984812i −0.836195 − 0.548432i \(-0.815225\pi\)
0.893053 + 0.449951i \(0.148558\pi\)
\(38\) 3.39291 − 5.87669i 0.550403 − 0.953325i
\(39\) 1.44450 − 6.44886i 0.231306 − 1.03264i
\(40\) 0.223904 − 5.41036i 0.0354023 − 0.855452i
\(41\) 10.6022 − 6.12120i 1.65579 − 0.955971i 0.681164 − 0.732131i \(-0.261474\pi\)
0.974626 − 0.223840i \(-0.0718594\pi\)
\(42\) −4.88330 − 15.6070i −0.753509 − 2.40821i
\(43\) 1.99045 − 3.44756i 0.303541 − 0.525748i −0.673395 − 0.739283i \(-0.735165\pi\)
0.976935 + 0.213535i \(0.0684979\pi\)
\(44\) −4.94910 − 8.57209i −0.746105 − 1.29229i
\(45\) −6.70242 + 0.278477i −0.999138 + 0.0415129i
\(46\) − 3.65123i − 0.538345i
\(47\) 5.63789 0.822371 0.411185 − 0.911552i \(-0.365115\pi\)
0.411185 + 0.911552i \(0.365115\pi\)
\(48\) 0.885103 + 0.814621i 0.127754 + 0.117580i
\(49\) 5.28245 + 9.14947i 0.754635 + 1.30707i
\(50\) 11.2253 + 0.930699i 1.58750 + 0.131621i
\(51\) −0.577853 − 1.84681i −0.0809157 − 0.258606i
\(52\) 5.86629 + 10.1607i 0.813507 + 1.40904i
\(53\) 3.70980 − 2.14185i 0.509580 − 0.294206i −0.223081 − 0.974800i \(-0.571611\pi\)
0.732661 + 0.680594i \(0.238278\pi\)
\(54\) −7.18687 + 9.23975i −0.978009 + 1.25737i
\(55\) 6.37697 − 3.33808i 0.859870 − 0.450106i
\(56\) 8.78953 + 5.07464i 1.17455 + 0.678127i
\(57\) 4.97926 − 1.55797i 0.659519 − 0.206358i
\(58\) −2.29622 −0.301509
\(59\) −6.43120 − 3.71306i −0.837271 − 0.483399i 0.0190644 − 0.999818i \(-0.493931\pi\)
−0.856336 + 0.516419i \(0.827265\pi\)
\(60\) 8.42046 − 8.42184i 1.08708 − 1.08725i
\(61\) − 12.0124i − 1.53803i −0.639230 − 0.769016i \(-0.720747\pi\)
0.639230 − 0.769016i \(-0.279253\pi\)
\(62\) −7.50138 + 10.0525i −0.952676 + 1.27667i
\(63\) 5.36351 − 11.3718i 0.675739 − 1.43271i
\(64\) −13.0464 −1.63081
\(65\) −7.55877 + 3.95671i −0.937550 + 0.490769i
\(66\) 2.74535 − 12.2564i 0.337930 − 1.50866i
\(67\) 12.2696 − 7.08384i 1.49897 − 0.865429i 0.498968 − 0.866620i \(-0.333712\pi\)
0.999999 + 0.00119117i \(0.000379162\pi\)
\(68\) 2.97519 + 1.71773i 0.360795 + 0.208305i
\(69\) 1.90109 − 2.06557i 0.228864 − 0.248666i
\(70\) −11.3028 + 17.8312i −1.35095 + 2.13124i
\(71\) −9.38271 + 5.41711i −1.11352 + 0.642893i −0.939739 − 0.341891i \(-0.888933\pi\)
−0.173783 + 0.984784i \(0.555599\pi\)
\(72\) −0.601471 − 7.24003i −0.0708841 − 0.853245i
\(73\) 1.40116 + 2.42688i 0.163994 + 0.284045i 0.936297 − 0.351208i \(-0.114229\pi\)
−0.772304 + 0.635253i \(0.780896\pi\)
\(74\) −0.779130 + 1.34949i −0.0905721 + 0.156875i
\(75\) 5.86580 + 6.37122i 0.677324 + 0.735685i
\(76\) −4.63123 + 8.02152i −0.531238 + 0.920132i
\(77\) 13.4908i 1.53742i
\(78\) −3.25413 + 14.5278i −0.368458 + 1.64495i
\(79\) 13.8006 + 7.96777i 1.55269 + 0.896445i 0.997922 + 0.0644383i \(0.0205256\pi\)
0.554766 + 0.832006i \(0.312808\pi\)
\(80\) 0.0642133 − 1.55163i 0.00717926 − 0.173478i
\(81\) −8.87662 + 1.48512i −0.986291 + 0.165013i
\(82\) −23.8844 + 13.7897i −2.63759 + 1.52281i
\(83\) −2.00628 + 1.15833i −0.220218 + 0.127143i −0.606051 − 0.795426i \(-0.707247\pi\)
0.385833 + 0.922569i \(0.373914\pi\)
\(84\) 6.66557 + 21.3031i 0.727273 + 2.32436i
\(85\) −1.33750 + 2.11001i −0.145072 + 0.228863i
\(86\) −4.48402 + 7.76655i −0.483524 + 0.837489i
\(87\) −1.29902 − 1.19558i −0.139269 − 0.128179i
\(88\) 3.89760 + 6.75085i 0.415486 + 0.719643i
\(89\) 16.0855 1.70506 0.852530 − 0.522678i \(-0.175067\pi\)
0.852530 + 0.522678i \(0.175067\pi\)
\(90\) 15.0990 − 0.627344i 1.59158 − 0.0661278i
\(91\) − 15.9910i − 1.67631i
\(92\) 4.98383i 0.519601i
\(93\) −9.47773 + 1.78117i −0.982795 + 0.184698i
\(94\) −12.7009 −1.30999
\(95\) −5.68888 − 3.60607i −0.583667 − 0.369975i
\(96\) −8.16642 − 7.51612i −0.833482 − 0.767111i
\(97\) − 13.7208i − 1.39314i −0.717489 − 0.696569i \(-0.754709\pi\)
0.717489 − 0.696569i \(-0.245291\pi\)
\(98\) −11.9001 − 20.6116i −1.20209 − 2.08209i
\(99\) 7.93464 − 5.50425i 0.797461 − 0.553198i
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 465.2.t.d.119.4 yes 104
3.2 odd 2 inner 465.2.t.d.119.50 yes 104
5.4 even 2 inner 465.2.t.d.119.49 yes 104
15.14 odd 2 inner 465.2.t.d.119.3 ✓ 104
31.6 odd 6 inner 465.2.t.d.254.3 yes 104
93.68 even 6 inner 465.2.t.d.254.49 yes 104
155.99 odd 6 inner 465.2.t.d.254.50 yes 104
465.254 even 6 inner 465.2.t.d.254.4 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.3 ✓ 104 15.14 odd 2 inner
465.2.t.d.119.4 yes 104 1.1 even 1 trivial
465.2.t.d.119.49 yes 104 5.4 even 2 inner
465.2.t.d.119.50 yes 104 3.2 odd 2 inner
465.2.t.d.254.3 yes 104 31.6 odd 6 inner
465.2.t.d.254.4 yes 104 465.254 even 6 inner
465.2.t.d.254.49 yes 104 93.68 even 6 inner
465.2.t.d.254.50 yes 104 155.99 odd 6 inner