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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 119.36
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.36

$q$-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} +(2.13127 - 0.676520i) q^{5} +(1.92923 + 0.0379990i) q^{6} +(1.74935 + 1.00999i) q^{7} -3.07355 q^{8} +(2.99767 + 0.118133i) q^{9} +(2.37436 - 0.753681i) 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} +(3.71383 - 1.09885i) q^{15} -1.90634 q^{16} +(-0.447757 - 0.258512i) q^{17} +(3.33958 + 0.131607i) q^{18} +(-1.42743 + 2.47239i) q^{19} +(-1.61738 + 0.513397i) q^{20} +(2.99492 + 1.80868i) q^{21} +(-0.802452 - 1.38989i) q^{22} +7.25758i q^{23} +(-5.32251 - 0.104835i) q^{24} +(4.08464 - 2.88370i) q^{25} +(-1.10771 - 1.91861i) q^{26} +(5.18709 + 0.306819i) q^{27} +(-1.32754 - 0.766458i) q^{28} -6.96082 q^{29} +(4.13741 - 1.22418i) 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} +(-6.55056 + 2.07932i) 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} +(6.46878 - 1.77621i) q^{45} +8.08535i q^{46} +8.36312 q^{47} +(-3.30124 - 0.0650228i) q^{48} +(-1.45985 - 2.52854i) q^{49} +(4.55052 - 3.21260i) 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} +(-2.37917 - 2.17166i) q^{55} +(-5.37670 - 3.10424i) q^{56} +(-2.55624 + 4.23279i) q^{57} -7.75474 q^{58} +(-10.0708 - 5.81440i) q^{59} +(-2.81835 + 0.833891i) q^{60} +9.17896i q^{61} +(-2.46449 - 5.69219i) q^{62} +(5.12466 + 3.23426i) q^{63} +8.29489 q^{64} +(-3.28423 - 2.99778i) 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} +(7.17179 - 4.85442i) q^{75} +(1.08325 - 1.87625i) q^{76} -2.90996i q^{77} +(-1.85280 - 3.36027i) q^{78} +(3.68045 + 2.12491i) q^{79} +(-4.06293 + 1.28968i) 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} +(7.20658 - 1.97880i) 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 \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.11406 0.787756 0.393878 − 0.919163i \(-0.371133\pi\)
0.393878 + 0.919163i \(0.371133\pi\)
\(3\) 1.73171 + 0.0341087i 0.999806 + 0.0196926i
\(4\) −0.758880 −0.379440
\(5\) 2.13127 − 0.676520i 0.953134 − 0.302549i
\(6\) 1.92923 + 0.0379990i 0.787604 + 0.0155130i
\(7\) 1.74935 + 1.00999i 0.661191 + 0.381739i 0.792731 − 0.609572i \(-0.208659\pi\)
−0.131539 + 0.991311i \(0.541992\pi\)
\(8\) −3.07355 −1.08666
\(9\) 2.99767 + 0.118133i 0.999224 + 0.0393777i
\(10\) 2.37436 − 0.753681i 0.750837 − 0.238335i
\(11\) −0.720298 − 1.24759i −0.217178 − 0.376163i 0.736766 − 0.676148i \(-0.236352\pi\)
−0.953944 + 0.299984i \(0.903019\pi\)
\(12\) −1.31416 − 0.0258844i −0.379366 − 0.00747217i
\(13\) −0.994305 − 1.72219i −0.275771 − 0.477649i 0.694559 − 0.719436i \(-0.255600\pi\)
−0.970329 + 0.241787i \(0.922266\pi\)
\(14\) 1.94887 + 1.12518i 0.520858 + 0.300717i
\(15\) 3.71383 − 1.09885i 0.958907 − 0.283721i
\(16\) −1.90634 −0.476586
\(17\) −0.447757 − 0.258512i −0.108597 − 0.0626985i 0.444718 − 0.895671i \(-0.353304\pi\)
−0.553315 + 0.832972i \(0.686637\pi\)
\(18\) 3.33958 + 0.131607i 0.787145 + 0.0310200i
\(19\) −1.42743 + 2.47239i −0.327476 + 0.567205i −0.982010 − 0.188827i \(-0.939531\pi\)
0.654534 + 0.756032i \(0.272865\pi\)
\(20\) −1.61738 + 0.513397i −0.361657 + 0.114799i
\(21\) 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
\(25\) 4.08464 − 2.88370i 0.816928 − 0.576740i
\(26\) −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 \(0.723683\pi\)
\(30\) 4.13741 − 1.22418i 0.755385 − 0.223503i
\(31\) −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 −0.481260 − 0.876578i \(-0.659821\pi\)
0.999769 − 0.0215061i \(-0.00684612\pi\)
\(38\) −1.59024 + 2.75438i −0.257971 + 0.446819i
\(39\) −1.66311 − 3.01625i −0.266311 − 0.482987i
\(40\) −6.55056 + 2.07932i −1.03573 + 0.328769i
\(41\) −5.50654 + 3.17920i −0.859976 + 0.496508i −0.864004 − 0.503484i \(-0.832051\pi\)
0.00402805 + 0.999992i \(0.498718\pi\)
\(42\) 3.33651 + 2.01497i 0.514835 + 0.310916i
\(43\) −1.20627 + 2.08932i −0.183955 + 0.318619i −0.943224 − 0.332158i \(-0.892223\pi\)
0.759269 + 0.650777i \(0.225557\pi\)
\(44\) 0.546619 + 0.946772i 0.0824059 + 0.142731i
\(45\) 6.46878 − 1.77621i 0.964308 − 0.264782i
\(46\) 8.08535i 1.19212i
\(47\) 8.36312 1.21989 0.609944 − 0.792445i \(-0.291192\pi\)
0.609944 + 0.792445i \(0.291192\pi\)
\(48\) −3.30124 − 0.0650228i −0.476493 − 0.00938523i
\(49\) −1.45985 − 2.52854i −0.208551 − 0.361220i
\(50\) 4.55052 − 3.21260i 0.643540 − 0.454330i
\(51\) −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 \(-0.875187\pi\)
−0.131108 + 0.991368i \(0.541853\pi\)
\(54\) 5.77870 + 0.341814i 0.786382 + 0.0465150i
\(55\) −2.37917 − 2.17166i −0.320807 − 0.292827i
\(56\) −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 −0.328831 − 0.944389i \(-0.606655\pi\)
−0.982280 + 0.187418i \(0.939988\pi\)
\(60\) −2.81835 + 0.833891i −0.363847 + 0.107655i
\(61\) 9.17896i 1.17525i 0.809135 + 0.587623i \(0.199936\pi\)
−0.809135 + 0.587623i \(0.800064\pi\)
\(62\) −2.46449 − 5.69219i −0.312991 − 0.722909i
\(63\) 5.12466 + 3.23426i 0.645647 + 0.407479i
\(64\) 8.29489 1.03686
\(65\) −3.28423 − 2.99778i −0.407359 − 0.371829i
\(66\) −1.34221 − 2.43426i −0.165215 − 0.299637i
\(67\) 2.91305 − 1.68185i 0.355886 − 0.205471i −0.311389 − 0.950283i \(-0.600794\pi\)
0.667275 + 0.744812i \(0.267461\pi\)
\(68\) 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 −0.820505 − 0.571639i \(-0.806308\pi\)
0.0848011 + 0.996398i \(0.472975\pi\)
\(72\) −9.21349 − 0.363087i −1.08582 − 0.0427902i
\(73\) −2.44105 − 4.22802i −0.285703 − 0.494852i 0.687076 − 0.726585i \(-0.258894\pi\)
−0.972779 + 0.231733i \(0.925560\pi\)
\(74\) 3.51370 − 6.08590i 0.408459 − 0.707472i
\(75\) 7.17179 − 4.85442i 0.828127 − 0.560540i
\(76\) 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 0.692542 − 0.721377i \(-0.256491\pi\)
−0.278460 + 0.960448i \(0.589824\pi\)
\(80\) −4.06293 + 1.28968i −0.454250 + 0.144191i
\(81\) 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 \(-0.902754\pi\)
−0.216366 + 0.976312i \(0.569421\pi\)
\(84\) −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 \(0.561844\pi\)
\(90\) 7.20658 − 1.97880i 0.759640 − 0.208584i
\(91\) − 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.119.36 yes 104
3.2 odd 2 inner 465.2.t.d.119.18 yes 104
5.4 even 2 inner 465.2.t.d.119.17 ✓ 104
15.14 odd 2 inner 465.2.t.d.119.35 yes 104
31.6 odd 6 inner 465.2.t.d.254.35 yes 104
93.68 even 6 inner 465.2.t.d.254.17 yes 104
155.99 odd 6 inner 465.2.t.d.254.18 yes 104
465.254 even 6 inner 465.2.t.d.254.36 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.17 ✓ 104 5.4 even 2 inner
465.2.t.d.119.18 yes 104 3.2 odd 2 inner
465.2.t.d.119.35 yes 104 15.14 odd 2 inner
465.2.t.d.119.36 yes 104 1.1 even 1 trivial
465.2.t.d.254.17 yes 104 93.68 even 6 inner
465.2.t.d.254.18 yes 104 155.99 odd 6 inner
465.2.t.d.254.35 yes 104 31.6 odd 6 inner
465.2.t.d.254.36 yes 104 465.254 even 6 inner