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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 119.20
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.20

$q$-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-0.784965 q^{2} +(-0.462641 - 1.66912i) q^{3} -1.38383 q^{4} +(2.07460 + 0.834291i) q^{5} +(0.363157 + 1.31020i) q^{6} +(-2.60048 - 1.50139i) q^{7} +2.65619 q^{8} +(-2.57193 + 1.54441i) q^{9} +(-1.62849 - 0.654889i) q^{10} +(1.16251 + 2.01353i) q^{11} +(0.640216 + 2.30978i) q^{12} +(-2.92486 - 5.06601i) q^{13} +(2.04128 + 1.17854i) q^{14} +(0.432738 - 3.84873i) q^{15} +0.682646 q^{16} +(-5.75273 - 3.32134i) q^{17} +(2.01887 - 1.21230i) q^{18} +(-1.52578 + 2.64272i) q^{19} +(-2.87089 - 1.15452i) q^{20} +(-1.30291 + 5.03511i) q^{21} +(-0.912530 - 1.58055i) q^{22} +6.91406i q^{23} +(-1.22886 - 4.43350i) q^{24} +(3.60792 + 3.46164i) q^{25} +(2.29591 + 3.97664i) q^{26} +(3.76768 + 3.57835i) q^{27} +(3.59862 + 2.07766i) q^{28} -3.05766 q^{29} +(-0.339684 + 3.02112i) q^{30} +(-5.37163 + 1.46477i) q^{31} -5.84823 q^{32} +(2.82299 - 2.87191i) q^{33} +(4.51569 + 2.60713i) q^{34} +(-4.14235 - 5.28433i) q^{35} +(3.55911 - 2.13720i) q^{36} +(-3.72056 + 6.44419i) q^{37} +(1.19768 - 2.07445i) q^{38} +(-7.10262 + 7.22569i) q^{39} +(5.51052 + 2.21603i) q^{40} +(1.74340 - 1.00656i) q^{41} +(1.02274 - 3.95239i) q^{42} +(1.23629 - 2.14132i) q^{43} +(-1.60872 - 2.78638i) q^{44} +(-6.62420 + 1.05829i) q^{45} -5.42730i q^{46} -1.52239 q^{47} +(-0.315820 - 1.13942i) q^{48} +(1.00832 + 1.74646i) q^{49} +(-2.83209 - 2.71726i) q^{50} +(-2.88227 + 11.1386i) q^{51} +(4.04751 + 7.01049i) q^{52} +(-6.30188 + 3.63839i) q^{53} +(-2.95750 - 2.80888i) q^{54} +(0.731876 + 5.14713i) q^{55} +(-6.90735 - 3.98796i) q^{56} +(5.11691 + 1.32407i) q^{57} +2.40015 q^{58} +(0.103101 + 0.0595252i) q^{59} +(-0.598835 + 5.32599i) q^{60} -7.50474i q^{61} +(4.21654 - 1.14980i) q^{62} +(9.00699 - 0.154738i) q^{63} +3.22536 q^{64} +(-1.84139 - 12.9501i) q^{65} +(-2.21595 + 2.25435i) q^{66} +(0.777063 - 0.448638i) q^{67} +(7.96079 + 4.59617i) q^{68} +(11.5404 - 3.19873i) q^{69} +(3.25160 + 4.14801i) q^{70} +(-0.857228 + 0.494921i) q^{71} +(-6.83152 + 4.10223i) q^{72} +(-1.33946 - 2.32001i) q^{73} +(2.92051 - 5.05847i) q^{74} +(4.10872 - 7.62354i) q^{75} +(2.11142 - 3.65708i) q^{76} -6.98151i q^{77} +(5.57531 - 5.67191i) q^{78} +(-13.7747 - 7.95282i) q^{79} +(1.41622 + 0.569525i) q^{80} +(4.22962 - 7.94420i) q^{81} +(-1.36851 + 0.790110i) q^{82} +(-2.81511 + 1.62530i) q^{83} +(1.80300 - 6.96774i) q^{84} +(-9.16363 - 11.6899i) q^{85} +(-0.970448 + 1.68086i) q^{86} +(1.41460 + 5.10360i) q^{87} +(3.08785 + 5.34831i) q^{88} +4.07108 q^{89} +(5.19976 - 0.830720i) q^{90} +17.5654i q^{91} -9.56789i q^{92} +(4.93002 + 8.28824i) q^{93} +1.19502 q^{94} +(-5.37018 + 4.20965i) q^{95} +(2.70563 + 9.76140i) q^{96} -19.0951i q^{97} +(-0.791495 - 1.37091i) q^{98} +(-6.09960 - 3.38326i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.784965 −0.555054 −0.277527 − 0.960718i \(-0.589515\pi\)
−0.277527 + 0.960718i \(0.589515\pi\)
\(3\) −0.462641 − 1.66912i −0.267106 − 0.963667i
\(4\) −1.38383 −0.691915
\(5\) 2.07460 + 0.834291i 0.927789 + 0.373106i
\(6\) 0.363157 + 1.31020i 0.148258 + 0.534887i
\(7\) −2.60048 − 1.50139i −0.982888 − 0.567470i −0.0797470 − 0.996815i \(-0.525411\pi\)
−0.903141 + 0.429345i \(0.858745\pi\)
\(8\) 2.65619 0.939104
\(9\) −2.57193 + 1.54441i −0.857309 + 0.514802i
\(10\) −1.62849 − 0.654889i −0.514973 − 0.207094i
\(11\) 1.16251 + 2.01353i 0.350510 + 0.607101i 0.986339 − 0.164729i \(-0.0526749\pi\)
−0.635829 + 0.771830i \(0.719342\pi\)
\(12\) 0.640216 + 2.30978i 0.184815 + 0.666776i
\(13\) −2.92486 − 5.06601i −0.811211 − 1.40506i −0.912017 − 0.410152i \(-0.865476\pi\)
0.100807 − 0.994906i \(-0.467858\pi\)
\(14\) 2.04128 + 1.17854i 0.545556 + 0.314977i
\(15\) 0.432738 − 3.84873i 0.111732 − 0.993738i
\(16\) 0.682646 0.170661
\(17\) −5.75273 − 3.32134i −1.39524 − 0.805543i −0.401351 − 0.915924i \(-0.631459\pi\)
−0.993889 + 0.110381i \(0.964793\pi\)
\(18\) 2.01887 − 1.21230i 0.475853 − 0.285743i
\(19\) −1.52578 + 2.64272i −0.350037 + 0.606283i −0.986256 − 0.165227i \(-0.947164\pi\)
0.636218 + 0.771509i \(0.280498\pi\)
\(20\) −2.87089 − 1.15452i −0.641951 − 0.258158i
\(21\) −1.30291 + 5.03511i −0.284318 + 1.09875i
\(22\) −0.912530 − 1.58055i −0.194552 − 0.336974i
\(23\) 6.91406i 1.44168i 0.693101 + 0.720841i \(0.256244\pi\)
−0.693101 + 0.720841i \(0.743756\pi\)
\(24\) −1.22886 − 4.43350i −0.250840 − 0.904984i
\(25\) 3.60792 + 3.46164i 0.721584 + 0.692327i
\(26\) 2.29591 + 3.97664i 0.450266 + 0.779883i
\(27\) 3.76768 + 3.57835i 0.725090 + 0.688654i
\(28\) 3.59862 + 2.07766i 0.680075 + 0.392641i
\(29\) −3.05766 −0.567793 −0.283896 − 0.958855i \(-0.591627\pi\)
−0.283896 + 0.958855i \(0.591627\pi\)
\(30\) −0.339684 + 3.02112i −0.0620175 + 0.551578i
\(31\) −5.37163 + 1.46477i −0.964774 + 0.263081i
\(32\) −5.84823 −1.03383
\(33\) 2.82299 − 2.87191i 0.491420 − 0.499935i
\(34\) 4.51569 + 2.60713i 0.774434 + 0.447120i
\(35\) −4.14235 − 5.28433i −0.700185 − 0.893214i
\(36\) 3.55911 − 2.13720i 0.593185 − 0.356199i
\(37\) −3.72056 + 6.44419i −0.611656 + 1.05942i 0.379306 + 0.925271i \(0.376163\pi\)
−0.990961 + 0.134147i \(0.957170\pi\)
\(38\) 1.19768 − 2.07445i 0.194290 − 0.336520i
\(39\) −7.10262 + 7.22569i −1.13733 + 1.15704i
\(40\) 5.51052 + 2.21603i 0.871290 + 0.350386i
\(41\) 1.74340 − 1.00656i 0.272274 − 0.157197i −0.357647 − 0.933857i \(-0.616421\pi\)
0.629921 + 0.776660i \(0.283087\pi\)
\(42\) 1.02274 − 3.95239i 0.157812 − 0.609866i
\(43\) 1.23629 − 2.14132i 0.188533 − 0.326549i −0.756228 − 0.654308i \(-0.772960\pi\)
0.944761 + 0.327759i \(0.106293\pi\)
\(44\) −1.60872 − 2.78638i −0.242523 − 0.420062i
\(45\) −6.62420 + 1.05829i −0.987477 + 0.157760i
\(46\) − 5.42730i − 0.800211i
\(47\) −1.52239 −0.222063 −0.111032 − 0.993817i \(-0.535416\pi\)
−0.111032 + 0.993817i \(0.535416\pi\)
\(48\) −0.315820 − 1.13942i −0.0455847 − 0.164461i
\(49\) 1.00832 + 1.74646i 0.144046 + 0.249494i
\(50\) −2.83209 − 2.71726i −0.400518 − 0.384279i
\(51\) −2.88227 + 11.1386i −0.403598 + 1.55971i
\(52\) 4.04751 + 7.01049i 0.561289 + 0.972181i
\(53\) −6.30188 + 3.63839i −0.865629 + 0.499771i −0.865893 − 0.500229i \(-0.833249\pi\)
0.000264094 1.00000i \(0.499916\pi\)
\(54\) −2.95750 − 2.80888i −0.402464 − 0.382240i
\(55\) 0.731876 + 5.14713i 0.0986861 + 0.694039i
\(56\) −6.90735 − 3.98796i −0.923034 − 0.532914i
\(57\) 5.11691 + 1.32407i 0.677752 + 0.175378i
\(58\) 2.40015 0.315156
\(59\) 0.103101 + 0.0595252i 0.0134226 + 0.00774952i 0.506696 − 0.862125i \(-0.330867\pi\)
−0.493274 + 0.869874i \(0.664200\pi\)
\(60\) −0.598835 + 5.32599i −0.0773093 + 0.687582i
\(61\) − 7.50474i − 0.960884i −0.877027 − 0.480442i \(-0.840476\pi\)
0.877027 − 0.480442i \(-0.159524\pi\)
\(62\) 4.21654 − 1.14980i 0.535502 − 0.146024i
\(63\) 9.00699 − 0.154738i 1.13477 − 0.0194952i
\(64\) 3.22536 0.403170
\(65\) −1.84139 − 12.9501i −0.228396 − 1.60626i
\(66\) −2.21595 + 2.25435i −0.272765 + 0.277491i
\(67\) 0.777063 − 0.448638i 0.0949334 − 0.0548098i −0.451782 − 0.892129i \(-0.649211\pi\)
0.546715 + 0.837319i \(0.315878\pi\)
\(68\) 7.96079 + 4.59617i 0.965388 + 0.557367i
\(69\) 11.5404 − 3.19873i 1.38930 − 0.385081i
\(70\) 3.25160 + 4.14801i 0.388641 + 0.495782i
\(71\) −0.857228 + 0.494921i −0.101734 + 0.0587363i −0.550004 − 0.835162i \(-0.685374\pi\)
0.448270 + 0.893898i \(0.352040\pi\)
\(72\) −6.83152 + 4.10223i −0.805103 + 0.483453i
\(73\) −1.33946 − 2.32001i −0.156772 − 0.271537i 0.776931 − 0.629586i \(-0.216775\pi\)
−0.933703 + 0.358049i \(0.883442\pi\)
\(74\) 2.92051 − 5.05847i 0.339502 − 0.588035i
\(75\) 4.10872 − 7.62354i 0.474434 − 0.880291i
\(76\) 2.11142 − 3.65708i 0.242196 − 0.419496i
\(77\) − 6.98151i − 0.795616i
\(78\) 5.57531 − 5.67191i 0.631279 − 0.642218i
\(79\) −13.7747 − 7.95282i −1.54977 − 0.894762i −0.998158 − 0.0606625i \(-0.980679\pi\)
−0.551614 − 0.834099i \(-0.685988\pi\)
\(80\) 1.41622 + 0.569525i 0.158338 + 0.0636748i
\(81\) 4.22962 − 7.94420i 0.469957 − 0.882689i
\(82\) −1.36851 + 0.790110i −0.151127 + 0.0872531i
\(83\) −2.81511 + 1.62530i −0.308998 + 0.178400i −0.646478 − 0.762933i \(-0.723759\pi\)
0.337480 + 0.941333i \(0.390425\pi\)
\(84\) 1.80300 − 6.96774i 0.196724 − 0.760243i
\(85\) −9.16363 − 11.6899i −0.993936 − 1.26795i
\(86\) −0.970448 + 1.68086i −0.104646 + 0.181252i
\(87\) 1.41460 + 5.10360i 0.151661 + 0.547163i
\(88\) 3.08785 + 5.34831i 0.329165 + 0.570131i
\(89\) 4.07108 0.431534 0.215767 − 0.976445i \(-0.430775\pi\)
0.215767 + 0.976445i \(0.430775\pi\)
\(90\) 5.19976 − 0.830720i 0.548103 − 0.0875655i
\(91\) 17.5654i 1.84135i
\(92\) − 9.56789i − 0.997521i
\(93\) 4.93002 + 8.28824i 0.511219 + 0.859450i
\(94\) 1.19502 0.123257
\(95\) −5.37018 + 4.20965i −0.550969 + 0.431901i
\(96\) 2.70563 + 9.76140i 0.276142 + 0.996269i
\(97\) − 19.0951i − 1.93881i −0.245467 − 0.969405i \(-0.578941\pi\)
0.245467 − 0.969405i \(-0.421059\pi\)
\(98\) −0.791495 − 1.37091i −0.0799530 − 0.138483i
\(99\) −6.09960 − 3.38326i −0.613032 − 0.340030i
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.119.20 yes 104
3.2 odd 2 inner 465.2.t.d.119.34 yes 104
5.4 even 2 inner 465.2.t.d.119.33 yes 104
15.14 odd 2 inner 465.2.t.d.119.19 ✓ 104
31.6 odd 6 inner 465.2.t.d.254.19 yes 104
93.68 even 6 inner 465.2.t.d.254.33 yes 104
155.99 odd 6 inner 465.2.t.d.254.34 yes 104
465.254 even 6 inner 465.2.t.d.254.20 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.19 ✓ 104 15.14 odd 2 inner
465.2.t.d.119.20 yes 104 1.1 even 1 trivial
465.2.t.d.119.33 yes 104 5.4 even 2 inner
465.2.t.d.119.34 yes 104 3.2 odd 2 inner
465.2.t.d.254.19 yes 104 31.6 odd 6 inner
465.2.t.d.254.20 yes 104 465.254 even 6 inner
465.2.t.d.254.33 yes 104 93.68 even 6 inner
465.2.t.d.254.34 yes 104 155.99 odd 6 inner