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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 254.14
Character \(\chi\) \(=\) 465.254
Dual form 465.2.t.d.119.14

$q$-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 \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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 −0.750509 − 0.660860i \(-0.770192\pi\)
0.197067 + 0.980390i \(0.436858\pi\)
\(8\) 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 0.233096 + 0.972454i \(0.425114\pi\)
−0.958718 + 0.284360i \(0.908219\pi\)
\(12\) −0.210979 − 0.110885i −0.0609043 − 0.0320098i
\(13\) −0.307395 + 0.532425i −0.0852562 + 0.147668i −0.905500 − 0.424346i \(-0.860504\pi\)
0.820244 + 0.572014i \(0.193838\pi\)
\(14\) 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 0.0327716 − 0.999463i \(-0.489567\pi\)
0.881946 + 0.471350i \(0.156233\pi\)
\(18\) −2.32182 − 3.37205i −0.547258 − 0.794799i
\(19\) 2.90894 + 5.03844i 0.667357 + 1.15590i 0.978640 + 0.205579i \(0.0659079\pi\)
−0.311283 + 0.950317i \(0.600759\pi\)
\(20\) −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 \(-0.847160\pi\)
0.886921 − 0.461922i \(-0.152840\pi\)
\(24\) 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 0.999090 − 0.0426578i \(-0.0135825\pi\)
−0.536488 + 0.843908i \(0.680249\pi\)
\(38\) −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 −0.287971 − 0.957639i \(-0.592981\pi\)
−0.973325 + 0.229429i \(0.926314\pi\)
\(42\) 3.99342 − 0.158623i 0.616198 − 0.0244761i
\(43\) 2.53588 + 4.39228i 0.386718 + 0.669816i 0.992006 − 0.126191i \(-0.0402752\pi\)
−0.605288 + 0.796007i \(0.706942\pi\)
\(44\) 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 0.810632 − 0.585556i \(-0.199124\pi\)
−0.101790 + 0.994806i \(0.532457\pi\)
\(54\) −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 −0.143066 − 0.989713i \(-0.545696\pi\)
0.785584 + 0.618755i \(0.212363\pi\)
\(60\) −0.525342 + 0.0897419i −0.0678213 + 0.0115856i
\(61\) − 9.69632i − 1.24149i −0.784014 − 0.620743i \(-0.786831\pi\)
0.784014 − 0.620743i \(-0.213169\pi\)
\(62\) 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 0.223620 − 0.974676i \(-0.428212\pi\)
−0.732284 + 0.680999i \(0.761546\pi\)
\(68\) −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 0.822068 − 0.569389i \(-0.192820\pi\)
−0.0820718 + 0.996626i \(0.526154\pi\)
\(72\) 4.96314 + 7.20811i 0.584911 + 0.849484i
\(73\) 0.805479 − 1.39513i 0.0942742 − 0.163288i −0.815031 − 0.579417i \(-0.803280\pi\)
0.909305 + 0.416129i \(0.136614\pi\)
\(74\) −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 −0.488346 − 0.872650i \(-0.662400\pi\)
0.511564 + 0.859245i \(0.329066\pi\)
\(80\) −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 0.595893 − 0.803064i \(-0.296798\pi\)
−0.397527 + 0.917590i \(0.630132\pi\)
\(84\) 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 \(-0.825723\pi\)
0.853824 − 0.520561i \(-0.174277\pi\)
\(98\) 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.254.14 yes 104
3.2 odd 2 inner 465.2.t.d.254.40 yes 104
5.4 even 2 inner 465.2.t.d.254.39 yes 104
15.14 odd 2 inner 465.2.t.d.254.13 yes 104
31.26 odd 6 inner 465.2.t.d.119.13 ✓ 104
93.26 even 6 inner 465.2.t.d.119.39 yes 104
155.119 odd 6 inner 465.2.t.d.119.40 yes 104
465.119 even 6 inner 465.2.t.d.119.14 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.13 ✓ 104 31.26 odd 6 inner
465.2.t.d.119.14 yes 104 465.119 even 6 inner
465.2.t.d.119.39 yes 104 93.26 even 6 inner
465.2.t.d.119.40 yes 104 155.119 odd 6 inner
465.2.t.d.254.13 yes 104 15.14 odd 2 inner
465.2.t.d.254.14 yes 104 1.1 even 1 trivial
465.2.t.d.254.39 yes 104 5.4 even 2 inner
465.2.t.d.254.40 yes 104 3.2 odd 2 inner