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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 119.14
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.14

$q$-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-1.36470 q^{2} +(1.53319 - 0.805808i) q^{3} -0.137607 q^{4} +(1.77591 + 1.35873i) q^{5} +(-2.09234 + 1.09968i) q^{6} +(-1.46427 - 0.845397i) q^{7} +2.91718 q^{8} +(1.70135 - 2.47091i) q^{9} +(-2.42357 - 1.85426i) q^{10} +(-2.40661 - 4.16838i) q^{11} +(-0.210979 + 0.110885i) q^{12} +(-0.307395 - 0.532425i) q^{13} +(1.99828 + 1.15371i) q^{14} +(3.81768 + 0.652159i) q^{15} -3.70585 q^{16} +(3.77148 + 2.17746i) q^{17} +(-2.32182 + 3.37205i) q^{18} +(2.90894 - 5.03844i) q^{19} +(-0.244378 - 0.186972i) q^{20} +(-2.92623 - 0.116234i) q^{21} +(3.28429 + 5.68857i) q^{22} +4.43060i q^{23} +(4.47260 - 2.35069i) q^{24} +(1.30768 + 4.82597i) q^{25} +(0.419501 + 0.726597i) q^{26} +(0.617405 - 5.15934i) q^{27} +(0.201495 + 0.116333i) q^{28} +8.46868 q^{29} +(-5.20997 - 0.889998i) q^{30} +(-0.0385902 - 5.56763i) q^{31} -0.777011 q^{32} +(-7.04871 - 4.45165i) q^{33} +(-5.14692 - 2.97157i) q^{34} +(-1.45174 - 3.49090i) q^{35} +(-0.234118 + 0.340016i) q^{36} +(2.81390 - 4.87382i) q^{37} +(-3.96982 + 6.87593i) q^{38} +(-0.900328 - 0.568607i) q^{39} +(5.18064 + 3.96367i) q^{40} +(-8.07624 + 4.66282i) q^{41} +(3.99342 + 0.158623i) q^{42} +(2.53588 - 4.39228i) q^{43} +(0.331168 + 0.573600i) q^{44} +(6.37875 - 2.07643i) q^{45} -6.04641i q^{46} -5.15494 q^{47} +(-5.68177 + 2.98620i) q^{48} +(-2.07061 - 3.58640i) q^{49} +(-1.78459 - 6.58597i) q^{50} +(7.53701 + 0.299379i) q^{51} +(0.0422999 + 0.0732656i) q^{52} +(5.16045 - 2.97938i) q^{53} +(-0.842570 + 7.04093i) q^{54} +(1.38980 - 10.6726i) q^{55} +(-4.27155 - 2.46618i) q^{56} +(0.399950 - 10.0689i) q^{57} -11.5572 q^{58} +(4.93527 + 2.84938i) q^{59} +(-0.525342 - 0.0897419i) q^{60} +9.69632i q^{61} +(0.0526639 + 7.59812i) q^{62} +(-4.58014 + 2.17978i) q^{63} +8.47208 q^{64} +(0.177518 - 1.36320i) q^{65} +(9.61934 + 6.07514i) q^{66} +(-4.16359 + 2.40385i) q^{67} +(-0.518984 - 0.299635i) q^{68} +(3.57021 + 6.79295i) q^{69} +(1.98118 + 4.76402i) q^{70} +(6.23532 - 3.59996i) q^{71} +(4.96314 - 7.20811i) q^{72} +(0.805479 + 1.39513i) q^{73} +(-3.84011 + 6.65127i) q^{74} +(5.89373 + 6.34539i) q^{75} +(-0.400292 + 0.693327i) q^{76} +8.13818i q^{77} +(1.22867 + 0.775974i) q^{78} +(0.206366 + 0.119146i) q^{79} +(-6.58124 - 5.03526i) q^{80} +(-3.21084 - 8.40776i) q^{81} +(11.0216 - 6.36332i) q^{82} +(1.80720 - 1.04339i) q^{83} +(0.402672 + 0.0159946i) q^{84} +(3.73920 + 8.99141i) q^{85} +(-3.46071 + 5.99412i) q^{86} +(12.9841 - 6.82413i) q^{87} +(-7.02053 - 12.1599i) q^{88} -14.1440 q^{89} +(-8.70504 + 2.83370i) q^{90} +1.03949i q^{91} -0.609683i q^{92} +(-4.54561 - 8.50514i) q^{93} +7.03493 q^{94} +(12.0119 - 4.99531i) q^{95} +(-1.19131 + 0.626122i) q^{96} +10.2539i q^{97} +(2.82575 + 4.89434i) q^{98} +(-14.3942 - 1.14532i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.36470 −0.964985 −0.482493 − 0.875900i \(-0.660268\pi\)
−0.482493 + 0.875900i \(0.660268\pi\)
\(3\) 1.53319 − 0.805808i 0.885188 − 0.465234i
\(4\) −0.137607 −0.0688037
\(5\) 1.77591 + 1.35873i 0.794209 + 0.607644i
\(6\) −2.09234 + 1.09968i −0.854193 + 0.448943i
\(7\) −1.46427 − 0.845397i −0.553443 − 0.319530i 0.197067 − 0.980390i \(-0.436858\pi\)
−0.750509 + 0.660860i \(0.770192\pi\)
\(8\) 2.91718 1.03138
\(9\) 1.70135 − 2.47091i 0.567116 − 0.823638i
\(10\) −2.42357 − 1.85426i −0.766400 − 0.586368i
\(11\) −2.40661 − 4.16838i −0.725622 − 1.25681i −0.958718 − 0.284360i \(-0.908219\pi\)
0.233096 − 0.972454i \(-0.425114\pi\)
\(12\) −0.210979 + 0.110885i −0.0609043 + 0.0320098i
\(13\) −0.307395 − 0.532425i −0.0852562 − 0.147668i 0.820244 − 0.572014i \(-0.193838\pi\)
−0.905500 + 0.424346i \(0.860504\pi\)
\(14\) 1.99828 + 1.15371i 0.534064 + 0.308342i
\(15\) 3.81768 + 0.652159i 0.985721 + 0.168387i
\(16\) −3.70585 −0.926462
\(17\) 3.77148 + 2.17746i 0.914718 + 0.528112i 0.881946 − 0.471350i \(-0.156233\pi\)
0.0327716 + 0.999463i \(0.489567\pi\)
\(18\) −2.32182 + 3.37205i −0.547258 + 0.794799i
\(19\) 2.90894 − 5.03844i 0.667357 − 1.15590i −0.311283 − 0.950317i \(-0.600759\pi\)
0.978640 − 0.205579i \(-0.0659079\pi\)
\(20\) −0.244378 − 0.186972i −0.0546446 − 0.0418082i
\(21\) −2.92623 − 0.116234i −0.638557 − 0.0253643i
\(22\) 3.28429 + 5.68857i 0.700214 + 1.21281i
\(23\) 4.43060i 0.923843i 0.886921 + 0.461922i \(0.152840\pi\)
−0.886921 + 0.461922i \(0.847160\pi\)
\(24\) 4.47260 − 2.35069i 0.912965 − 0.479832i
\(25\) 1.30768 + 4.82597i 0.261537 + 0.965194i
\(26\) 0.419501 + 0.726597i 0.0822709 + 0.142497i
\(27\) 0.617405 − 5.15934i 0.118820 − 0.992916i
\(28\) 0.201495 + 0.116333i 0.0380789 + 0.0219849i
\(29\) 8.46868 1.57260 0.786298 − 0.617848i \(-0.211995\pi\)
0.786298 + 0.617848i \(0.211995\pi\)
\(30\) −5.20997 − 0.889998i −0.951206 − 0.162491i
\(31\) −0.0385902 − 5.56763i −0.00693101 − 0.999976i
\(32\) −0.777011 −0.137357
\(33\) −7.04871 − 4.45165i −1.22702 − 0.774933i
\(34\) −5.14692 − 2.97157i −0.882689 − 0.509621i
\(35\) −1.45174 − 3.49090i −0.245388 − 0.590070i
\(36\) −0.234118 + 0.340016i −0.0390197 + 0.0566694i
\(37\) 2.81390 − 4.87382i 0.462602 − 0.801250i −0.536488 − 0.843908i \(-0.680249\pi\)
0.999090 + 0.0426578i \(0.0135825\pi\)
\(38\) −3.96982 + 6.87593i −0.643990 + 1.11542i
\(39\) −0.900328 − 0.568607i −0.144168 − 0.0910499i
\(40\) 5.18064 + 3.96367i 0.819131 + 0.626712i
\(41\) −8.07624 + 4.66282i −1.26130 + 0.728210i −0.973325 − 0.229429i \(-0.926314\pi\)
−0.287971 + 0.957639i \(0.592981\pi\)
\(42\) 3.99342 + 0.158623i 0.616198 + 0.0244761i
\(43\) 2.53588 − 4.39228i 0.386718 − 0.669816i −0.605288 − 0.796007i \(-0.706942\pi\)
0.992006 + 0.126191i \(0.0402752\pi\)
\(44\) 0.331168 + 0.573600i 0.0499255 + 0.0864735i
\(45\) 6.37875 − 2.07643i 0.950888 − 0.309537i
\(46\) − 6.04641i − 0.891495i
\(47\) −5.15494 −0.751926 −0.375963 − 0.926635i \(-0.622688\pi\)
−0.375963 + 0.926635i \(0.622688\pi\)
\(48\) −5.68177 + 2.98620i −0.820093 + 0.431021i
\(49\) −2.07061 − 3.58640i −0.295801 − 0.512342i
\(50\) −1.78459 − 6.58597i −0.252379 − 0.931397i
\(51\) 7.53701 + 0.299379i 1.05539 + 0.0419215i
\(52\) 0.0422999 + 0.0732656i 0.00586594 + 0.0101601i
\(53\) 5.16045 − 2.97938i 0.708842 − 0.409250i −0.101790 − 0.994806i \(-0.532457\pi\)
0.810632 + 0.585556i \(0.199124\pi\)
\(54\) −0.842570 + 7.04093i −0.114659 + 0.958149i
\(55\) 1.38980 − 10.6726i 0.187400 − 1.43909i
\(56\) −4.27155 − 2.46618i −0.570809 − 0.329557i
\(57\) 0.399950 − 10.0689i 0.0529747 − 1.33366i
\(58\) −11.5572 −1.51753
\(59\) 4.93527 + 2.84938i 0.642518 + 0.370958i 0.785584 − 0.618755i \(-0.212363\pi\)
−0.143066 + 0.989713i \(0.545696\pi\)
\(60\) −0.525342 − 0.0897419i −0.0678213 − 0.0115856i
\(61\) 9.69632i 1.24149i 0.784014 + 0.620743i \(0.213169\pi\)
−0.784014 + 0.620743i \(0.786831\pi\)
\(62\) 0.0526639 + 7.59812i 0.00668832 + 0.964962i
\(63\) −4.58014 + 2.17978i −0.577043 + 0.274626i
\(64\) 8.47208 1.05901
\(65\) 0.177518 − 1.36320i 0.0220184 − 0.169085i
\(66\) 9.61934 + 6.07514i 1.18406 + 0.747798i
\(67\) −4.16359 + 2.40385i −0.508664 + 0.293677i −0.732284 − 0.680999i \(-0.761546\pi\)
0.223620 + 0.974676i \(0.428212\pi\)
\(68\) −0.518984 − 0.299635i −0.0629360 − 0.0363361i
\(69\) 3.57021 + 6.79295i 0.429803 + 0.817775i
\(70\) 1.98118 + 4.76402i 0.236796 + 0.569409i
\(71\) 6.23532 − 3.59996i 0.739996 − 0.427237i −0.0820718 − 0.996626i \(-0.526154\pi\)
0.822068 + 0.569389i \(0.192820\pi\)
\(72\) 4.96314 − 7.20811i 0.584911 − 0.849484i
\(73\) 0.805479 + 1.39513i 0.0942742 + 0.163288i 0.909305 − 0.416129i \(-0.136614\pi\)
−0.815031 + 0.579417i \(0.803280\pi\)
\(74\) −3.84011 + 6.65127i −0.446404 + 0.773195i
\(75\) 5.89373 + 6.34539i 0.680550 + 0.732702i
\(76\) −0.400292 + 0.693327i −0.0459167 + 0.0795300i
\(77\) 8.13818i 0.927432i
\(78\) 1.22867 + 0.775974i 0.139120 + 0.0878618i
\(79\) 0.206366 + 0.119146i 0.0232180 + 0.0134049i 0.511564 − 0.859245i \(-0.329066\pi\)
−0.488346 + 0.872650i \(0.662400\pi\)
\(80\) −6.58124 − 5.03526i −0.735805 − 0.562960i
\(81\) −3.21084 − 8.40776i −0.356760 − 0.934196i
\(82\) 11.0216 − 6.36332i 1.21713 − 0.702712i
\(83\) 1.80720 − 1.04339i 0.198366 − 0.114527i −0.397527 − 0.917590i \(-0.630132\pi\)
0.595893 + 0.803064i \(0.296798\pi\)
\(84\) 0.402672 + 0.0159946i 0.0439351 + 0.00174516i
\(85\) 3.73920 + 8.99141i 0.405573 + 0.975255i
\(86\) −3.46071 + 5.99412i −0.373177 + 0.646362i
\(87\) 12.9841 − 6.82413i 1.39204 − 0.731624i
\(88\) −7.02053 − 12.1599i −0.748391 − 1.29625i
\(89\) −14.1440 −1.49926 −0.749631 − 0.661856i \(-0.769769\pi\)
−0.749631 + 0.661856i \(0.769769\pi\)
\(90\) −8.70504 + 2.83370i −0.917592 + 0.298698i
\(91\) 1.03949i 0.108968i
\(92\) − 0.609683i − 0.0635639i
\(93\) −4.54561 − 8.50514i −0.471358 − 0.881942i
\(94\) 7.03493 0.725597
\(95\) 12.0119 − 4.99531i 1.23240 − 0.512508i
\(96\) −1.19131 + 0.626122i −0.121587 + 0.0639033i
\(97\) 10.2539i 1.04112i 0.853824 + 0.520561i \(0.174277\pi\)
−0.853824 + 0.520561i \(0.825723\pi\)
\(98\) 2.82575 + 4.89434i 0.285443 + 0.494403i
\(99\) −14.3942 − 1.14532i −1.44667 − 0.115109i
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.119.14 yes 104
3.2 odd 2 inner 465.2.t.d.119.40 yes 104
5.4 even 2 inner 465.2.t.d.119.39 yes 104
15.14 odd 2 inner 465.2.t.d.119.13 ✓ 104
31.6 odd 6 inner 465.2.t.d.254.13 yes 104
93.68 even 6 inner 465.2.t.d.254.39 yes 104
155.99 odd 6 inner 465.2.t.d.254.40 yes 104
465.254 even 6 inner 465.2.t.d.254.14 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.13 ✓ 104 15.14 odd 2 inner
465.2.t.d.119.14 yes 104 1.1 even 1 trivial
465.2.t.d.119.39 yes 104 5.4 even 2 inner
465.2.t.d.119.40 yes 104 3.2 odd 2 inner
465.2.t.d.254.13 yes 104 31.6 odd 6 inner
465.2.t.d.254.14 yes 104 465.254 even 6 inner
465.2.t.d.254.39 yes 104 93.68 even 6 inner
465.2.t.d.254.40 yes 104 155.99 odd 6 inner