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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 119.39
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.39

$q$-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} +(-0.288745 - 2.21735i) q^{5} +(-2.09234 + 1.09968i) q^{6} +(1.46427 + 0.845397i) q^{7} -2.91718 q^{8} +(1.70135 - 2.47091i) q^{9} +(-0.394049 - 3.02600i) 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} +(2.22946 + 3.16694i) q^{15} -3.70585 q^{16} +(-3.77148 - 2.17746i) q^{17} +(2.32182 - 3.37205i) q^{18} +(2.90894 - 5.03844i) q^{19} +(0.0397335 + 0.305124i) q^{20} +(-2.92623 - 0.116234i) q^{21} +(-3.28429 - 5.68857i) q^{22} -4.43060i q^{23} +(4.47260 - 2.35069i) q^{24} +(-4.83325 + 1.28050i) q^{25} +(0.419501 + 0.726597i) q^{26} +(-0.617405 + 5.15934i) q^{27} +(-0.201495 - 0.116333i) q^{28} +8.46868 q^{29} +(3.04253 + 4.32191i) 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} +(0.842323 + 6.46840i) 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} +(-5.97013 - 3.05901i) q^{45} -6.04641i q^{46} +5.15494 q^{47} +(5.68177 - 2.98620i) q^{48} +(-2.07061 - 3.58640i) q^{49} +(-6.59592 + 1.74749i) 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} +(-8.54784 + 6.53990i) q^{55} +(-4.27155 - 2.46618i) q^{56} +(-0.399950 + 10.0689i) q^{57} +11.5572 q^{58} +(4.93527 + 2.84938i) q^{59} +(-0.306790 - 0.435795i) q^{60} +9.69632i q^{61} +(-0.0526639 - 7.59812i) q^{62} +(4.58014 - 2.17978i) q^{63} +8.47208 q^{64} +(1.09181 - 0.835337i) 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} +(6.37846 - 5.85792i) q^{75} +(-0.400292 + 0.693327i) q^{76} -8.13818i q^{77} +(-1.22867 - 0.775974i) q^{78} +(0.206366 + 0.119146i) q^{79} +(1.07005 + 8.21715i) 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} +(-8.14741 - 4.17462i) 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 \(-0.339732\pi\)
0.482493 + 0.875900i \(0.339732\pi\)
\(3\) −1.53319 + 0.805808i −0.885188 + 0.465234i
\(4\) −0.137607 −0.0688037
\(5\) −0.288745 − 2.21735i −0.129131 − 0.991628i
\(6\) −2.09234 + 1.09968i −0.854193 + 0.448943i
\(7\) 1.46427 + 0.845397i 0.553443 + 0.319530i 0.750509 − 0.660860i \(-0.229808\pi\)
−0.197067 + 0.980390i \(0.563142\pi\)
\(8\) −2.91718 −1.03138
\(9\) 1.70135 − 2.47091i 0.567116 − 0.823638i
\(10\) −0.394049 − 3.02600i −0.124609 − 0.956906i
\(11\) −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 0.905500 − 0.424346i \(-0.139496\pi\)
−0.820244 + 0.572014i \(0.806162\pi\)
\(14\) 1.99828 + 1.15371i 0.534064 + 0.308342i
\(15\) 2.22946 + 3.16694i 0.575643 + 0.817701i
\(16\) −3.70585 −0.926462
\(17\) −3.77148 − 2.17746i −0.914718 − 0.528112i −0.0327716 − 0.999463i \(-0.510433\pi\)
−0.881946 + 0.471350i \(0.843767\pi\)
\(18\) 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 \(-0.0659079\pi\)
\(20\) 0.0397335 + 0.305124i 0.00888468 + 0.0682277i
\(21\) −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
\(25\) −4.83325 + 1.28050i −0.966650 + 0.256099i
\(26\) 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 \(0.211995\pi\)
\(30\) 3.04253 + 4.32191i 0.555487 + 0.789069i
\(31\) −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 \(-0.986417\pi\)
0.536488 + 0.843908i \(0.319751\pi\)
\(38\) 3.96982 − 6.87593i 0.643990 − 1.11542i
\(39\) −0.900328 − 0.568607i −0.144168 − 0.0910499i
\(40\) 0.842323 + 6.46840i 0.133183 + 1.02274i
\(41\) −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 −0.992006 − 0.126191i \(-0.959725\pi\)
0.605288 + 0.796007i \(0.293058\pi\)
\(44\) 0.331168 + 0.573600i 0.0499255 + 0.0864735i
\(45\) −5.97013 − 3.05901i −0.889974 − 0.456010i
\(46\) − 6.04641i − 0.891495i
\(47\) 5.15494 0.751926 0.375963 − 0.926635i \(-0.377312\pi\)
0.375963 + 0.926635i \(0.377312\pi\)
\(48\) 5.68177 − 2.98620i 0.820093 − 0.431021i
\(49\) −2.07061 − 3.58640i −0.295801 − 0.512342i
\(50\) −6.59592 + 1.74749i −0.932803 + 0.247132i
\(51\) 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 \(-0.800876\pi\)
0.101790 + 0.994806i \(0.467543\pi\)
\(54\) −0.842570 + 7.04093i −0.114659 + 0.958149i
\(55\) −8.54784 + 6.53990i −1.15259 + 0.881840i
\(56\) −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 \(0.545696\pi\)
\(60\) −0.306790 − 0.435795i −0.0396064 − 0.0562609i
\(61\) 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
\(65\) 1.09181 − 0.835337i 0.135422 − 0.103611i
\(66\) 9.61934 + 6.07514i 1.18406 + 0.747798i
\(67\) 4.16359 − 2.40385i 0.508664 − 0.293677i −0.223620 − 0.974676i \(-0.571788\pi\)
0.732284 + 0.680999i \(0.238454\pi\)
\(68\) 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 0.815031 − 0.579417i \(-0.196720\pi\)
−0.909305 + 0.416129i \(0.863386\pi\)
\(74\) −3.84011 + 6.65127i −0.446404 + 0.773195i
\(75\) 6.37846 − 5.85792i 0.736521 − 0.676414i
\(76\) −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 \(0.662400\pi\)
\(80\) 1.07005 + 8.21715i 0.119635 + 0.918706i
\(81\) −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 \(-0.703202\pi\)
0.397527 + 0.917590i \(0.369868\pi\)
\(84\) 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 \(0.769769\pi\)
\(90\) −8.14741 − 4.17462i −0.858812 − 0.440043i
\(91\) 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.119.39 yes 104
3.2 odd 2 inner 465.2.t.d.119.13 ✓ 104
5.4 even 2 inner 465.2.t.d.119.14 yes 104
15.14 odd 2 inner 465.2.t.d.119.40 yes 104
31.6 odd 6 inner 465.2.t.d.254.40 yes 104
93.68 even 6 inner 465.2.t.d.254.14 yes 104
155.99 odd 6 inner 465.2.t.d.254.13 yes 104
465.254 even 6 inner 465.2.t.d.254.39 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.13 ✓ 104 3.2 odd 2 inner
465.2.t.d.119.14 yes 104 5.4 even 2 inner
465.2.t.d.119.39 yes 104 1.1 even 1 trivial
465.2.t.d.119.40 yes 104 15.14 odd 2 inner
465.2.t.d.254.13 yes 104 155.99 odd 6 inner
465.2.t.d.254.14 yes 104 93.68 even 6 inner
465.2.t.d.254.39 yes 104 465.254 even 6 inner
465.2.t.d.254.40 yes 104 31.6 odd 6 inner