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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 254.39
Character \(\chi\) \(=\) 465.254
Dual form 465.2.t.d.119.39

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

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

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 \(-0.339732\pi\)
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 −0.197067 − 0.980390i \(-0.563142\pi\)
0.750509 + 0.660860i \(0.229808\pi\)
\(8\) −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 0.233096 + 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 −0.820244 − 0.572014i \(-0.806162\pi\)
0.905500 + 0.424346i \(0.139496\pi\)
\(14\) 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 −0.881946 − 0.471350i \(-0.843767\pi\)
−0.0327716 + 0.999463i \(0.510433\pi\)
\(18\) 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 \(0.600759\pi\)
\(20\) 0.0397335 − 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 \(0.152840\pi\)
−0.886921 + 0.461922i \(0.847160\pi\)
\(24\) 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 0.536488 − 0.843908i \(-0.319751\pi\)
−0.999090 + 0.0426578i \(0.986417\pi\)
\(38\) 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 −0.287971 − 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 0.605288 − 0.796007i \(-0.293058\pi\)
−0.992006 + 0.126191i \(0.959725\pi\)
\(44\) 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 0.101790 − 0.994806i \(-0.467543\pi\)
−0.810632 + 0.585556i \(0.800876\pi\)
\(54\) −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 −0.143066 − 0.989713i \(-0.545696\pi\)
0.785584 + 0.618755i \(0.212363\pi\)
\(60\) −0.306790 + 0.435795i −0.0396064 + 0.0562609i
\(61\) − 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
\(65\) 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 0.732284 − 0.680999i \(-0.238454\pi\)
−0.223620 + 0.974676i \(0.571788\pi\)
\(68\) 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 −0.909305 − 0.416129i \(-0.863386\pi\)
0.815031 + 0.579417i \(0.196720\pi\)
\(74\) −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 −0.488346 − 0.872650i \(-0.662400\pi\)
0.511564 + 0.859245i \(0.329066\pi\)
\(80\) 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 0.397527 − 0.917590i \(-0.369868\pi\)
−0.595893 + 0.803064i \(0.703202\pi\)
\(84\) 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 \(0.174277\pi\)
−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.254.39 yes 104
3.2 odd 2 inner 465.2.t.d.254.13 yes 104
5.4 even 2 inner 465.2.t.d.254.14 yes 104
15.14 odd 2 inner 465.2.t.d.254.40 yes 104
31.26 odd 6 inner 465.2.t.d.119.40 yes 104
93.26 even 6 inner 465.2.t.d.119.14 yes 104
155.119 odd 6 inner 465.2.t.d.119.13 ✓ 104
465.119 even 6 inner 465.2.t.d.119.39 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.13 ✓ 104 155.119 odd 6 inner
465.2.t.d.119.14 yes 104 93.26 even 6 inner
465.2.t.d.119.39 yes 104 465.119 even 6 inner
465.2.t.d.119.40 yes 104 31.26 odd 6 inner
465.2.t.d.254.13 yes 104 3.2 odd 2 inner
465.2.t.d.254.14 yes 104 5.4 even 2 inner
465.2.t.d.254.39 yes 104 1.1 even 1 trivial
465.2.t.d.254.40 yes 104 15.14 odd 2 inner