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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 254.17
Character \(\chi\) \(=\) 465.254
Dual form 465.2.t.d.119.17

$q$-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-1.11406 q^{2} +(-1.73171 + 0.0341087i) q^{3} -0.758880 q^{4} +(1.65152 + 1.50748i) q^{5} +(1.92923 - 0.0379990i) q^{6} +(-1.74935 + 1.00999i) q^{7} +3.07355 q^{8} +(2.99767 - 0.118133i) q^{9} +(-1.83989 - 1.67941i) q^{10} +(-0.720298 + 1.24759i) q^{11} +(1.31416 - 0.0258844i) q^{12} +(0.994305 - 1.72219i) q^{13} +(1.94887 - 1.12518i) q^{14} +(-2.91138 - 2.55419i) q^{15} -1.90634 q^{16} +(0.447757 - 0.258512i) q^{17} +(-3.33958 + 0.131607i) q^{18} +(-1.42743 - 2.47239i) q^{19} +(-1.25330 - 1.14399i) q^{20} +(2.99492 - 1.80868i) q^{21} +(0.802452 - 1.38989i) q^{22} +7.25758i q^{23} +(-5.32251 + 0.104835i) q^{24} +(0.455035 + 4.97925i) q^{25} +(-1.10771 + 1.91861i) q^{26} +(-5.18709 + 0.306819i) q^{27} +(1.32754 - 0.766458i) q^{28} -6.96082 q^{29} +(3.24344 + 2.84551i) q^{30} +(-2.21218 + 5.10943i) q^{31} -4.02332 q^{32} +(1.20480 - 2.18504i) q^{33} +(-0.498826 + 0.287997i) q^{34} +(-4.41161 - 0.969086i) q^{35} +(-2.27487 + 0.0896487i) q^{36} +(-3.15397 - 5.46283i) q^{37} +(1.59024 + 2.75438i) q^{38} +(-1.66311 + 3.01625i) q^{39} +(5.07602 + 4.63330i) q^{40} +(-5.50654 - 3.17920i) q^{41} +(-3.33651 + 2.01497i) q^{42} +(1.20627 + 2.08932i) q^{43} +(0.546619 - 0.946772i) q^{44} +(5.12880 + 4.32382i) q^{45} -8.08535i q^{46} -8.36312 q^{47} +(3.30124 - 0.0650228i) q^{48} +(-1.45985 + 2.52854i) q^{49} +(-0.506934 - 5.54716i) q^{50} +(-0.766570 + 0.462942i) q^{51} +(-0.754558 + 1.30693i) q^{52} +(7.68206 + 4.43524i) q^{53} +(5.77870 - 0.341814i) q^{54} +(-3.07030 + 0.974592i) q^{55} +(-5.37670 + 3.10424i) q^{56} +(2.55624 + 4.23279i) q^{57} +7.75474 q^{58} +(-10.0708 + 5.81440i) q^{59} +(2.20939 + 1.93832i) q^{60} -9.17896i q^{61} +(2.46449 - 5.69219i) q^{62} +(-5.12466 + 3.23426i) q^{63} +8.29489 q^{64} +(4.23827 - 1.34534i) q^{65} +(-1.34221 + 2.43426i) q^{66} +(-2.91305 - 1.68185i) q^{67} +(-0.339793 + 0.196180i) q^{68} +(-0.247547 - 12.5681i) q^{69} +(4.91478 + 1.07962i) q^{70} +(-6.19915 - 3.57908i) q^{71} +(9.21349 - 0.363087i) q^{72} +(2.44105 - 4.22802i) q^{73} +(3.51370 + 6.08590i) q^{74} +(-0.957827 - 8.60712i) q^{75} +(1.08325 + 1.87625i) q^{76} -2.90996i q^{77} +(1.85280 - 3.36027i) q^{78} +(3.68045 - 2.12491i) q^{79} +(-3.14836 - 2.87376i) q^{80} +(8.97209 - 0.708248i) q^{81} +(6.13459 + 3.54181i) q^{82} +(10.6598 + 6.15441i) q^{83} +(-2.27279 + 1.37257i) q^{84} +(1.12918 + 0.248044i) q^{85} +(-1.34385 - 2.32762i) q^{86} +(12.0542 - 0.237424i) q^{87} +(-2.21387 + 3.83453i) q^{88} -3.64281 q^{89} +(-5.71377 - 4.81698i) q^{90} +4.01694i q^{91} -5.50763i q^{92} +(3.65659 - 8.92353i) q^{93} +9.31699 q^{94} +(1.36963 - 6.23502i) q^{95} +(6.96724 - 0.137230i) q^{96} +9.33547i q^{97} +(1.62636 - 2.81694i) q^{98} +(-2.01184 + 3.82496i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.11406 −0.787756 −0.393878 − 0.919163i \(-0.628867\pi\)
−0.393878 + 0.919163i \(0.628867\pi\)
\(3\) −1.73171 + 0.0341087i −0.999806 + 0.0196926i
\(4\) −0.758880 −0.379440
\(5\) 1.65152 + 1.50748i 0.738582 + 0.674164i
\(6\) 1.92923 − 0.0379990i 0.787604 − 0.0155130i
\(7\) −1.74935 + 1.00999i −0.661191 + 0.381739i −0.792731 − 0.609572i \(-0.791341\pi\)
0.131539 + 0.991311i \(0.458008\pi\)
\(8\) 3.07355 1.08666
\(9\) 2.99767 − 0.118133i 0.999224 − 0.0393777i
\(10\) −1.83989 − 1.67941i −0.581823 − 0.531077i
\(11\) −0.720298 + 1.24759i −0.217178 + 0.376163i −0.953944 − 0.299984i \(-0.903019\pi\)
0.736766 + 0.676148i \(0.236352\pi\)
\(12\) 1.31416 − 0.0258844i 0.379366 − 0.00747217i
\(13\) 0.994305 − 1.72219i 0.275771 − 0.477649i −0.694559 − 0.719436i \(-0.744400\pi\)
0.970329 + 0.241787i \(0.0777336\pi\)
\(14\) 1.94887 − 1.12518i 0.520858 − 0.300717i
\(15\) −2.91138 − 2.55419i −0.751715 − 0.659488i
\(16\) −1.90634 −0.476586
\(17\) 0.447757 − 0.258512i 0.108597 − 0.0626985i −0.444718 − 0.895671i \(-0.646696\pi\)
0.553315 + 0.832972i \(0.313363\pi\)
\(18\) −3.33958 + 0.131607i −0.787145 + 0.0310200i
\(19\) −1.42743 − 2.47239i −0.327476 − 0.567205i 0.654534 − 0.756032i \(-0.272865\pi\)
−0.982010 + 0.188827i \(0.939531\pi\)
\(20\) −1.25330 − 1.14399i −0.280247 − 0.255804i
\(21\) 2.99492 − 1.80868i 0.653546 − 0.394686i
\(22\) 0.802452 − 1.38989i 0.171083 − 0.296325i
\(23\) 7.25758i 1.51331i 0.653814 + 0.756656i \(0.273168\pi\)
−0.653814 + 0.756656i \(0.726832\pi\)
\(24\) −5.32251 + 0.104835i −1.08645 + 0.0213993i
\(25\) 0.455035 + 4.97925i 0.0910070 + 0.995850i
\(26\) −1.10771 + 1.91861i −0.217240 + 0.376271i
\(27\) −5.18709 + 0.306819i −0.998255 + 0.0590474i
\(28\) 1.32754 − 0.766458i 0.250882 − 0.144847i
\(29\) −6.96082 −1.29259 −0.646296 − 0.763087i \(-0.723683\pi\)
−0.646296 + 0.763087i \(0.723683\pi\)
\(30\) 3.24344 + 2.84551i 0.592168 + 0.519516i
\(31\) −2.21218 + 5.10943i −0.397319 + 0.917680i
\(32\) −4.02332 −0.711229
\(33\) 1.20480 − 2.18504i 0.209728 − 0.380367i
\(34\) −0.498826 + 0.287997i −0.0855480 + 0.0493911i
\(35\) −4.41161 − 0.969086i −0.745699 − 0.163806i
\(36\) −2.27487 + 0.0896487i −0.379146 + 0.0149415i
\(37\) −3.15397 − 5.46283i −0.518509 − 0.898084i −0.999769 − 0.0215061i \(-0.993154\pi\)
0.481260 − 0.876578i \(-0.340179\pi\)
\(38\) 1.59024 + 2.75438i 0.257971 + 0.446819i
\(39\) −1.66311 + 3.01625i −0.266311 + 0.482987i
\(40\) 5.07602 + 4.63330i 0.802590 + 0.732588i
\(41\) −5.50654 − 3.17920i −0.859976 − 0.496508i 0.00402805 − 0.999992i \(-0.498718\pi\)
−0.864004 + 0.503484i \(0.832051\pi\)
\(42\) −3.33651 + 2.01497i −0.514835 + 0.310916i
\(43\) 1.20627 + 2.08932i 0.183955 + 0.318619i 0.943224 − 0.332158i \(-0.107777\pi\)
−0.759269 + 0.650777i \(0.774443\pi\)
\(44\) 0.546619 − 0.946772i 0.0824059 − 0.142731i
\(45\) 5.12880 + 4.32382i 0.764556 + 0.644557i
\(46\) − 8.08535i − 1.19212i
\(47\) −8.36312 −1.21989 −0.609944 − 0.792445i \(-0.708808\pi\)
−0.609944 + 0.792445i \(0.708808\pi\)
\(48\) 3.30124 − 0.0650228i 0.476493 − 0.00938523i
\(49\) −1.45985 + 2.52854i −0.208551 + 0.361220i
\(50\) −0.506934 − 5.54716i −0.0716914 − 0.784487i
\(51\) −0.766570 + 0.462942i −0.107341 + 0.0648249i
\(52\) −0.754558 + 1.30693i −0.104638 + 0.181239i
\(53\) 7.68206 + 4.43524i 1.05521 + 0.609227i 0.924104 − 0.382141i \(-0.124813\pi\)
0.131108 + 0.991368i \(0.458147\pi\)
\(54\) 5.77870 − 0.341814i 0.786382 − 0.0465150i
\(55\) −3.07030 + 0.974592i −0.413999 + 0.131414i
\(56\) −5.37670 + 3.10424i −0.718492 + 0.414821i
\(57\) 2.55624 + 4.23279i 0.338582 + 0.560646i
\(58\) 7.75474 1.01825
\(59\) −10.0708 + 5.81440i −1.31111 + 0.756971i −0.982280 − 0.187418i \(-0.939988\pi\)
−0.328831 + 0.944389i \(0.606655\pi\)
\(60\) 2.20939 + 1.93832i 0.285231 + 0.250236i
\(61\) − 9.17896i − 1.17525i −0.809135 − 0.587623i \(-0.800064\pi\)
0.809135 − 0.587623i \(-0.199936\pi\)
\(62\) 2.46449 − 5.69219i 0.312991 − 0.722909i
\(63\) −5.12466 + 3.23426i −0.645647 + 0.407479i
\(64\) 8.29489 1.03686
\(65\) 4.23827 − 1.34534i 0.525693 − 0.166868i
\(66\) −1.34221 + 2.43426i −0.165215 + 0.299637i
\(67\) −2.91305 − 1.68185i −0.355886 − 0.205471i 0.311389 − 0.950283i \(-0.399206\pi\)
−0.667275 + 0.744812i \(0.732539\pi\)
\(68\) −0.339793 + 0.196180i −0.0412060 + 0.0237903i
\(69\) −0.247547 − 12.5681i −0.0298011 − 1.51302i
\(70\) 4.91478 + 1.07962i 0.587429 + 0.129039i
\(71\) −6.19915 − 3.57908i −0.735704 − 0.424759i 0.0848011 − 0.996398i \(-0.472975\pi\)
−0.820505 + 0.571639i \(0.806308\pi\)
\(72\) 9.21349 − 0.363087i 1.08582 − 0.0427902i
\(73\) 2.44105 − 4.22802i 0.285703 − 0.494852i −0.687076 − 0.726585i \(-0.741106\pi\)
0.972779 + 0.231733i \(0.0744396\pi\)
\(74\) 3.51370 + 6.08590i 0.408459 + 0.707472i
\(75\) −0.957827 − 8.60712i −0.110600 − 0.993865i
\(76\) 1.08325 + 1.87625i 0.124257 + 0.215220i
\(77\) − 2.90996i − 0.331621i
\(78\) 1.85280 − 3.36027i 0.209788 − 0.380476i
\(79\) 3.68045 − 2.12491i 0.414083 − 0.239071i −0.278460 − 0.960448i \(-0.589824\pi\)
0.692542 + 0.721377i \(0.256491\pi\)
\(80\) −3.14836 − 2.87376i −0.351998 − 0.321297i
\(81\) 8.97209 − 0.708248i 0.996899 − 0.0786942i
\(82\) 6.13459 + 3.54181i 0.677452 + 0.391127i
\(83\) 10.6598 + 6.15441i 1.17006 + 0.675535i 0.953694 − 0.300778i \(-0.0972462\pi\)
0.216366 + 0.976312i \(0.430579\pi\)
\(84\) −2.27279 + 1.37257i −0.247981 + 0.149759i
\(85\) 1.12918 + 0.248044i 0.122477 + 0.0269041i
\(86\) −1.34385 − 2.32762i −0.144911 − 0.250994i
\(87\) 12.0542 − 0.237424i 1.29234 − 0.0254546i
\(88\) −2.21387 + 3.83453i −0.235999 + 0.408762i
\(89\) −3.64281 −0.386137 −0.193069 − 0.981185i \(-0.561844\pi\)
−0.193069 + 0.981185i \(0.561844\pi\)
\(90\) −5.71377 − 4.81698i −0.602284 − 0.507754i
\(91\) 4.01694i 0.421090i
\(92\) − 5.50763i − 0.574210i
\(93\) 3.65659 − 8.92353i 0.379171 − 0.925327i
\(94\) 9.31699 0.960974
\(95\) 1.36963 − 6.23502i 0.140521 − 0.639700i
\(96\) 6.96724 − 0.137230i 0.711091 − 0.0140060i
\(97\) 9.33547i 0.947874i 0.880559 + 0.473937i \(0.157167\pi\)
−0.880559 + 0.473937i \(0.842833\pi\)
\(98\) 1.62636 − 2.81694i 0.164287 − 0.284554i
\(99\) −2.01184 + 3.82496i −0.202197 + 0.384423i
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.254.17 yes 104
3.2 odd 2 inner 465.2.t.d.254.35 yes 104
5.4 even 2 inner 465.2.t.d.254.36 yes 104
15.14 odd 2 inner 465.2.t.d.254.18 yes 104
31.26 odd 6 inner 465.2.t.d.119.18 yes 104
93.26 even 6 inner 465.2.t.d.119.36 yes 104
155.119 odd 6 inner 465.2.t.d.119.35 yes 104
465.119 even 6 inner 465.2.t.d.119.17 ✓ 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.17 ✓ 104 465.119 even 6 inner
465.2.t.d.119.18 yes 104 31.26 odd 6 inner
465.2.t.d.119.35 yes 104 155.119 odd 6 inner
465.2.t.d.119.36 yes 104 93.26 even 6 inner
465.2.t.d.254.17 yes 104 1.1 even 1 trivial
465.2.t.d.254.18 yes 104 15.14 odd 2 inner
465.2.t.d.254.35 yes 104 3.2 odd 2 inner
465.2.t.d.254.36 yes 104 5.4 even 2 inner