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Properties

Label 465.2.t.d.119.3
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 119.3
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.3

$q$-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-2.25277 q^{2} +(-1.65302 - 0.517218i) q^{3} +3.07497 q^{4} +(1.98107 + 1.03701i) q^{5} +(3.72388 + 1.16517i) q^{6} +(3.62956 + 2.09552i) q^{7} -2.42166 q^{8} +(2.46497 + 1.70995i) q^{9} +(-4.46288 - 2.33613i) q^{10} +(1.60948 + 2.78770i) q^{11} +(-5.08299 - 1.59043i) 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} +(-5.55301 - 3.85212i) q^{18} +(-1.50611 + 2.60865i) q^{19} +(6.09171 + 3.18876i) q^{20} +(-4.91590 - 5.34122i) q^{21} +(-3.62579 - 6.28004i) q^{22} +1.62078i q^{23} +(4.00305 + 1.25252i) q^{24} +(2.84924 + 4.10875i) 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} +(7.57971 + 5.25803i) q^{36} +(-0.345854 + 0.599038i) q^{37} +(3.39291 - 5.87669i) q^{38} +(1.44450 + 6.44886i) q^{39} +(-4.79746 - 2.51127i) q^{40} +(-10.6022 + 6.12120i) q^{41} +(11.0744 + 12.0325i) q^{42} +(-1.99045 + 3.44756i) q^{43} +(4.94910 + 8.57209i) q^{44} +(3.11004 + 5.94371i) q^{45} -3.65123i q^{46} +5.63789 q^{47} +(1.14803 + 0.359211i) q^{48} +(5.28245 + 9.14947i) q^{49} +(-6.41867 - 9.25607i) q^{50} +(-1.31046 - 1.42384i) q^{51} +(-5.86629 - 10.1607i) q^{52} +(3.70980 - 2.14185i) q^{53} +(7.18687 + 9.23975i) q^{54} +(0.297622 + 7.19165i) q^{55} +(-8.78953 - 5.07464i) q^{56} +(3.83887 - 3.53318i) 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} +(-0.352778 - 8.52444i) q^{65} +(2.74535 + 12.2564i) q^{66} +(-12.2696 + 7.08384i) q^{67} +(2.97519 + 1.71773i) q^{68} +(0.838295 - 2.67918i) q^{69} +(-11.3028 - 17.8312i) q^{70} +(9.38271 - 5.41711i) q^{71} +(-5.96931 - 4.14090i) q^{72} +(-1.40116 - 2.42688i) q^{73} +(0.779130 - 1.34949i) q^{74} +(-2.58473 - 8.26554i) q^{75} +(-4.63123 + 8.02152i) q^{76} +13.4908i q^{77} +(-3.25413 - 14.5278i) q^{78} +(13.8006 + 7.96777i) q^{79} +(-1.37586 - 0.720206i) q^{80} +(3.15216 + 8.42994i) q^{81} +(23.8844 - 13.7897i) q^{82} +(-2.00628 + 1.15833i) q^{83} +(-15.1162 - 16.4241i) q^{84} +(1.33750 + 2.11001i) q^{85} +(4.48402 - 7.76655i) q^{86} +(1.68491 + 0.527195i) q^{87} +(-3.89760 - 6.75085i) q^{88} -16.0855 q^{89} +(-7.00621 - 13.3898i) q^{90} -15.9910i q^{91} +4.98383i q^{92} +(-7.81230 + 5.65403i) q^{93} -12.7009 q^{94} +(-5.68888 + 3.60607i) q^{95} +(-10.5924 - 3.31427i) q^{96} +13.7208i q^{97} +(-11.9001 - 20.6116i) q^{98} +(-0.799499 + 9.62372i) 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 \(0.793304\pi\)
\(3\) −1.65302 − 0.517218i −0.954373 − 0.298616i
\(4\) 3.07497 1.53748
\(5\) 1.98107 + 1.03701i 0.885959 + 0.463763i
\(6\) 3.72388 + 1.16517i 1.52027 + 0.475680i
\(7\) 3.62956 + 2.09552i 1.37184 + 0.792034i 0.991160 − 0.132672i \(-0.0423556\pi\)
0.380683 + 0.924706i \(0.375689\pi\)
\(8\) −2.42166 −0.856185
\(9\) 2.46497 + 1.70995i 0.821657 + 0.569982i
\(10\) −4.46288 − 2.33613i −1.41129 − 0.738751i
\(11\) 1.60948 + 2.78770i 0.485276 + 0.840523i 0.999857 − 0.0169189i \(-0.00538570\pi\)
−0.514581 + 0.857442i \(0.672052\pi\)
\(12\) −5.08299 − 1.59043i −1.46733 − 0.459117i
\(13\) −1.90775 − 3.30433i −0.529116 − 0.916456i −0.999423 − 0.0339531i \(-0.989190\pi\)
0.470308 − 0.882503i \(-0.344143\pi\)
\(14\) −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 \(0.623408\pi\)
\(18\) −5.55301 − 3.85212i −1.30886 − 0.907952i
\(19\) −1.50611 + 2.60865i −0.345524 + 0.598466i −0.985449 − 0.169972i \(-0.945632\pi\)
0.639924 + 0.768438i \(0.278966\pi\)
\(20\) 6.09171 + 3.18876i 1.36215 + 0.713029i
\(21\) −4.91590 − 5.34122i −1.07274 − 1.16555i
\(22\) −3.62579 − 6.28004i −0.773020 − 1.33891i
\(23\) 1.62078i 0.337955i 0.985620 + 0.168978i \(0.0540465\pi\)
−0.985620 + 0.168978i \(0.945953\pi\)
\(24\) 4.00305 + 1.25252i 0.817120 + 0.255670i
\(25\) 2.84924 + 4.10875i 0.569847 + 0.821750i
\(26\) 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 \(-0.530170\pi\)
−0.0946387 + 0.995512i \(0.530170\pi\)
\(30\) 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
\(36\) 7.57971 + 5.25803i 1.26328 + 0.876339i
\(37\) −0.345854 + 0.599038i −0.0568581 + 0.0984812i −0.893053 − 0.449951i \(-0.851442\pi\)
0.836195 + 0.548432i \(0.184775\pi\)
\(38\) 3.39291 − 5.87669i 0.550403 − 0.953325i
\(39\) 1.44450 + 6.44886i 0.231306 + 1.03264i
\(40\) −4.79746 − 2.51127i −0.758545 − 0.397067i
\(41\) −10.6022 + 6.12120i −1.65579 + 0.955971i −0.681164 + 0.732131i \(0.738526\pi\)
−0.974626 + 0.223840i \(0.928141\pi\)
\(42\) 11.0744 + 12.0325i 1.70881 + 1.85666i
\(43\) −1.99045 + 3.44756i −0.303541 + 0.525748i −0.976935 − 0.213535i \(-0.931502\pi\)
0.673395 + 0.739283i \(0.264835\pi\)
\(44\) 4.94910 + 8.57209i 0.746105 + 1.29229i
\(45\) 3.11004 + 5.94371i 0.463618 + 0.886035i
\(46\) − 3.65123i − 0.538345i
\(47\) 5.63789 0.822371 0.411185 − 0.911552i \(-0.365115\pi\)
0.411185 + 0.911552i \(0.365115\pi\)
\(48\) 1.14803 + 0.359211i 0.165704 + 0.0518476i
\(49\) 5.28245 + 9.14947i 0.754635 + 1.30707i
\(50\) −6.41867 − 9.25607i −0.907738 − 1.30901i
\(51\) −1.31046 − 1.42384i −0.183501 − 0.199378i
\(52\) −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
\(55\) 0.297622 + 7.19165i 0.0401313 + 0.969722i
\(56\) −8.78953 − 5.07464i −1.17455 − 0.678127i
\(57\) 3.83887 − 3.53318i 0.508471 − 0.467981i
\(58\) 2.29622 0.301509
\(59\) 6.43120 + 3.71306i 0.837271 + 0.483399i 0.856336 − 0.516419i \(-0.172735\pi\)
−0.0190644 + 0.999818i \(0.506069\pi\)
\(60\) −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
\(65\) −0.352778 − 8.52444i −0.0437567 − 1.05733i
\(66\) 2.74535 + 12.2564i 0.337930 + 1.50866i
\(67\) −12.2696 + 7.08384i −1.49897 + 0.865429i −0.999999 − 0.00119117i \(-0.999621\pi\)
−0.498968 + 0.866620i \(0.666288\pi\)
\(68\) 2.97519 + 1.71773i 0.360795 + 0.208305i
\(69\) 0.838295 − 2.67918i 0.100919 − 0.322535i
\(70\) −11.3028 − 17.8312i −1.35095 − 2.13124i
\(71\) 9.38271 − 5.41711i 1.11352 − 0.642893i 0.173783 − 0.984784i \(-0.444401\pi\)
0.939739 + 0.341891i \(0.111067\pi\)
\(72\) −5.96931 − 4.14090i −0.703490 − 0.488010i
\(73\) −1.40116 − 2.42688i −0.163994 − 0.284045i 0.772304 − 0.635253i \(-0.219104\pi\)
−0.936297 + 0.351208i \(0.885771\pi\)
\(74\) 0.779130 − 1.34949i 0.0905721 − 0.156875i
\(75\) −2.58473 − 8.26554i −0.298459 − 0.954422i
\(76\) −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 \(0.312808\pi\)
\(80\) −1.37586 − 0.720206i −0.153826 − 0.0805215i
\(81\) 3.15216 + 8.42994i 0.350240 + 0.936660i
\(82\) 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 \(0.373914\pi\)
\(84\) −15.1162 − 16.4241i −1.64932 − 1.79202i
\(85\) 1.33750 + 2.11001i 0.145072 + 0.228863i
\(86\) 4.48402 − 7.76655i 0.483524 − 0.837489i
\(87\) 1.68491 + 0.527195i 0.180641 + 0.0565212i
\(88\) −3.89760 − 6.75085i −0.415486 − 0.719643i
\(89\) −16.0855 −1.70506 −0.852530 − 0.522678i \(-0.824933\pi\)
−0.852530 + 0.522678i \(0.824933\pi\)
\(90\) −7.00621 − 13.3898i −0.738519 − 1.41141i
\(91\) − 15.9910i − 1.67631i
\(92\) 4.98383i 0.519601i
\(93\) −7.81230 + 5.65403i −0.810097 + 0.586295i
\(94\) −12.7009 −1.30999
\(95\) −5.68888 + 3.60607i −0.583667 + 0.369975i
\(96\) −10.5924 − 3.31427i −1.08108 − 0.338261i
\(97\) 13.7208i 1.39314i 0.717489 + 0.696569i \(0.245291\pi\)
−0.717489 + 0.696569i \(0.754709\pi\)
\(98\) −11.9001 − 20.6116i −1.20209 − 2.08209i
\(99\) −0.799499 + 9.62372i −0.0803527 + 0.967221i
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.119.3 ✓ 104
3.2 odd 2 inner 465.2.t.d.119.49 yes 104
5.4 even 2 inner 465.2.t.d.119.50 yes 104
15.14 odd 2 inner 465.2.t.d.119.4 yes 104
31.6 odd 6 inner 465.2.t.d.254.4 yes 104
93.68 even 6 inner 465.2.t.d.254.50 yes 104
155.99 odd 6 inner 465.2.t.d.254.49 yes 104
465.254 even 6 inner 465.2.t.d.254.3 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.3 ✓ 104 1.1 even 1 trivial
465.2.t.d.119.4 yes 104 15.14 odd 2 inner
465.2.t.d.119.49 yes 104 3.2 odd 2 inner
465.2.t.d.119.50 yes 104 5.4 even 2 inner
465.2.t.d.254.3 yes 104 465.254 even 6 inner
465.2.t.d.254.4 yes 104 31.6 odd 6 inner
465.2.t.d.254.49 yes 104 155.99 odd 6 inner
465.2.t.d.254.50 yes 104 93.68 even 6 inner