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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 254.18
Character \(\chi\) \(=\) 465.254
Dual form 465.2.t.d.119.18

$q$-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} +(-0.895396 + 1.48265i) q^{3} -0.758880 q^{4} +(-2.13127 - 0.676520i) q^{5} +(0.997522 - 1.65176i) q^{6} +(1.74935 - 1.00999i) q^{7} +3.07355 q^{8} +(-1.39653 - 2.65513i) q^{9} +(2.37436 + 0.753681i) q^{10} +(0.720298 - 1.24759i) q^{11} +(0.679498 - 1.12516i) 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} +(1.55581 + 2.95796i) q^{18} +(-1.42743 - 2.47239i) q^{19} +(1.61738 + 0.513397i) q^{20} +(-0.0688986 + 3.49802i) q^{21} +(-0.802452 + 1.38989i) q^{22} +7.25758i q^{23} +(-2.75204 + 4.55701i) 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} +(-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} +(1.05980 + 2.01492i) 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} +(0.0767569 - 3.89699i) q^{42} +(-1.20627 - 2.08932i) q^{43} +(-0.546619 + 0.946772i) q^{44} +(1.18014 + 6.60358i) q^{45} -8.08535i q^{46} -8.36312 q^{47} +(1.70693 - 2.82645i) q^{48} +(-1.45985 + 2.52854i) q^{49} +(-4.55052 - 3.21260i) q^{50} +(-0.0176350 + 0.895340i) 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} +(4.94382 + 0.0973758i) 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} +(3.28423 - 2.99778i) q^{65} +(-1.34221 - 2.43426i) q^{66} +(2.91305 + 1.68185i) q^{67} +(-0.339793 + 0.196180i) q^{68} +(-10.7605 - 6.49842i) q^{69} +(4.91478 - 1.07962i) q^{70} +(6.19915 + 3.57908i) q^{71} +(-4.29230 - 8.16066i) q^{72} +(-2.44105 + 4.22802i) q^{73} +(-3.51370 - 6.08590i) q^{74} +(-7.93290 + 3.47406i) 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} +(-5.09941 + 7.41593i) q^{81} +(-6.13459 - 3.54181i) q^{82} +(10.6598 + 6.15441i) q^{83} +(0.0522857 - 2.65457i) q^{84} +(-1.12918 + 0.248044i) q^{85} +(1.34385 + 2.32762i) q^{86} +(-6.23269 + 10.3205i) q^{87} +(2.21387 - 3.83453i) q^{88} +3.64281 q^{89} +(-1.31474 - 7.35676i) q^{90} +4.01694i q^{91} -5.50763i q^{92} +(-5.59474 - 7.85486i) q^{93} +9.31699 q^{94} +(1.36963 + 6.23502i) q^{95} +(3.60247 - 5.96519i) q^{96} -9.33547i q^{97} +(1.62636 - 2.81694i) q^{98} +(-4.31843 - 0.170182i) 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 \(0.628867\pi\)
\(3\) −0.895396 + 1.48265i −0.516957 + 0.856011i
\(4\) −0.758880 −0.379440
\(5\) −2.13127 − 0.676520i −0.953134 − 0.302549i
\(6\) 0.997522 − 1.65176i 0.407236 − 0.674328i
\(7\) 1.74935 − 1.00999i 0.661191 − 0.381739i −0.131539 − 0.991311i \(-0.541992\pi\)
0.792731 + 0.609572i \(0.208659\pi\)
\(8\) 3.07355 1.08666
\(9\) −1.39653 − 2.65513i −0.465510 − 0.885043i
\(10\) 2.37436 + 0.753681i 0.750837 + 0.238335i
\(11\) 0.720298 − 1.24759i 0.217178 − 0.376163i −0.736766 − 0.676148i \(-0.763648\pi\)
0.953944 + 0.299984i \(0.0969815\pi\)
\(12\) 0.679498 − 1.12516i 0.196154 − 0.324805i
\(13\) −0.994305 + 1.72219i −0.275771 + 0.477649i −0.970329 − 0.241787i \(-0.922266\pi\)
0.694559 + 0.719436i \(0.255600\pi\)
\(14\) −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 \(0.313363\pi\)
\(18\) 1.55581 + 2.95796i 0.366709 + 0.697198i
\(19\) −1.42743 − 2.47239i −0.327476 − 0.567205i 0.654534 − 0.756032i \(-0.272865\pi\)
−0.982010 + 0.188827i \(0.939531\pi\)
\(20\) 1.61738 + 0.513397i 0.361657 + 0.114799i
\(21\) −0.0688986 + 3.49802i −0.0150349 + 0.763330i
\(22\) −0.802452 + 1.38989i −0.171083 + 0.296325i
\(23\) 7.25758i 1.51331i 0.653814 + 0.756656i \(0.273168\pi\)
−0.653814 + 0.756656i \(0.726832\pi\)
\(24\) −2.75204 + 4.55701i −0.561758 + 0.930195i
\(25\) 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 \(-0.276317\pi\)
0.646296 + 0.763087i \(0.276317\pi\)
\(30\) −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
\(36\) 1.05980 + 2.01492i 0.176633 + 0.335820i
\(37\) 3.15397 + 5.46283i 0.518509 + 0.898084i 0.999769 + 0.0215061i \(0.00684612\pi\)
−0.481260 + 0.876578i \(0.659821\pi\)
\(38\) 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 \(-0.167949\pi\)
−0.00402805 + 0.999992i \(0.501282\pi\)
\(42\) 0.0767569 − 3.89699i 0.0118438 − 0.601318i
\(43\) −1.20627 − 2.08932i −0.183955 − 0.318619i 0.759269 − 0.650777i \(-0.225557\pi\)
−0.943224 + 0.332158i \(0.892223\pi\)
\(44\) −0.546619 + 0.946772i −0.0824059 + 0.142731i
\(45\) 1.18014 + 6.60358i 0.175925 + 0.984404i
\(46\) − 8.08535i − 1.19212i
\(47\) −8.36312 −1.21989 −0.609944 − 0.792445i \(-0.708808\pi\)
−0.609944 + 0.792445i \(0.708808\pi\)
\(48\) 1.70693 − 2.82645i 0.246374 − 0.407963i
\(49\) −1.45985 + 2.52854i −0.208551 + 0.361220i
\(50\) −4.55052 − 3.21260i −0.643540 − 0.454330i
\(51\) −0.0176350 + 0.895340i −0.00246940 + 0.125373i
\(52\) 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
\(55\) −2.37917 + 2.17166i −0.320807 + 0.292827i
\(56\) 5.37670 − 3.10424i 0.718492 − 0.414821i
\(57\) 4.94382 + 0.0973758i 0.654825 + 0.0128977i
\(58\) −7.75474 −1.01825
\(59\) 10.0708 − 5.81440i 1.31111 − 0.756971i 0.328831 − 0.944389i \(-0.393345\pi\)
0.982280 + 0.187418i \(0.0600119\pi\)
\(60\) −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
\(65\) 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 0.667275 − 0.744812i \(-0.267461\pi\)
−0.311389 + 0.950283i \(0.600794\pi\)
\(68\) −0.339793 + 0.196180i −0.0412060 + 0.0237903i
\(69\) −10.7605 − 6.49842i −1.29541 − 0.782317i
\(70\) 4.91478 − 1.07962i 0.587429 − 0.129039i
\(71\) 6.19915 + 3.57908i 0.735704 + 0.424759i 0.820505 − 0.571639i \(-0.193692\pi\)
−0.0848011 + 0.996398i \(0.527025\pi\)
\(72\) −4.29230 − 8.16066i −0.505852 − 0.961743i
\(73\) −2.44105 + 4.22802i −0.285703 + 0.494852i −0.972779 − 0.231733i \(-0.925560\pi\)
0.687076 + 0.726585i \(0.258894\pi\)
\(74\) −3.51370 − 6.08590i −0.408459 − 0.707472i
\(75\) −7.93290 + 3.47406i −0.916012 + 0.401150i
\(76\) 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 \(0.256491\pi\)
\(80\) 4.06293 + 1.28968i 0.454250 + 0.144191i
\(81\) −5.09941 + 7.41593i −0.566601 + 0.823993i
\(82\) −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 \(0.430579\pi\)
\(84\) 0.0522857 − 2.65457i 0.00570484 − 0.289638i
\(85\) −1.12918 + 0.248044i −0.122477 + 0.0269041i
\(86\) 1.34385 + 2.32762i 0.144911 + 0.250994i
\(87\) −6.23269 + 10.3205i −0.668215 + 1.10647i
\(88\) 2.21387 − 3.83453i 0.235999 − 0.408762i
\(89\) 3.64281 0.386137 0.193069 − 0.981185i \(-0.438156\pi\)
0.193069 + 0.981185i \(0.438156\pi\)
\(90\) −1.31474 − 7.35676i −0.138586 − 0.775470i
\(91\) 4.01694i 0.421090i
\(92\) − 5.50763i − 0.574210i
\(93\) −5.59474 − 7.85486i −0.580148 − 0.814511i
\(94\) 9.31699 0.960974
\(95\) 1.36963 + 6.23502i 0.140521 + 0.639700i
\(96\) 3.60247 − 5.96519i 0.367675 − 0.608820i
\(97\) − 9.33547i − 0.947874i −0.880559 − 0.473937i \(-0.842833\pi\)
0.880559 − 0.473937i \(-0.157167\pi\)
\(98\) 1.62636 − 2.81694i 0.164287 − 0.284554i
\(99\) −4.31843 − 0.170182i −0.434019 − 0.0171039i
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.254.18 yes 104
3.2 odd 2 inner 465.2.t.d.254.36 yes 104
5.4 even 2 inner 465.2.t.d.254.35 yes 104
15.14 odd 2 inner 465.2.t.d.254.17 yes 104
31.26 odd 6 inner 465.2.t.d.119.17 ✓ 104
93.26 even 6 inner 465.2.t.d.119.35 yes 104
155.119 odd 6 inner 465.2.t.d.119.36 yes 104
465.119 even 6 inner 465.2.t.d.119.18 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.17 ✓ 104 31.26 odd 6 inner
465.2.t.d.119.18 yes 104 465.119 even 6 inner
465.2.t.d.119.35 yes 104 93.26 even 6 inner
465.2.t.d.119.36 yes 104 155.119 odd 6 inner
465.2.t.d.254.17 yes 104 15.14 odd 2 inner
465.2.t.d.254.18 yes 104 1.1 even 1 trivial
465.2.t.d.254.35 yes 104 5.4 even 2 inner
465.2.t.d.254.36 yes 104 3.2 odd 2 inner