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Properties

Label 465.2.t.d.119.18
Level $465$
Weight $2$
Character 465.119
Analytic conductor $3.713$
Analytic rank $0$
Dimension $104$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

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 119.18
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.18

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

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

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 \(-0.628867\pi\)
−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 0.792731 − 0.609572i \(-0.208659\pi\)
−0.131539 + 0.991311i \(0.541992\pi\)
\(8\) 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 0.953944 − 0.299984i \(-0.0969815\pi\)
−0.736766 + 0.676148i \(0.763648\pi\)
\(12\) 0.679498 + 1.12516i 0.196154 + 0.324805i
\(13\) −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
\(15\) 2.91138 + 2.55419i 0.751715 + 0.659488i
\(16\) −1.90634 −0.476586
\(17\) 0.447757 + 0.258512i 0.108597 + 0.0626985i 0.553315 − 0.832972i \(-0.313363\pi\)
−0.444718 + 0.895671i \(0.646696\pi\)
\(18\) 1.55581 − 2.95796i 0.366709 − 0.697198i
\(19\) −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
\(21\) −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 \(-0.726832\pi\)
0.653814 − 0.756656i \(-0.273168\pi\)
\(24\) −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 −0.481260 − 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 −0.00402805 − 0.999992i \(-0.501282\pi\)
0.864004 + 0.503484i \(0.167949\pi\)
\(42\) 0.0767569 + 3.89699i 0.0118438 + 0.601318i
\(43\) −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
\(45\) 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 0.131108 − 0.991368i \(-0.458147\pi\)
0.924104 + 0.382141i \(0.124813\pi\)
\(54\) −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 0.982280 − 0.187418i \(-0.0600119\pi\)
0.328831 + 0.944389i \(0.393345\pi\)
\(60\) −2.20939 − 1.93832i −0.285231 − 0.250236i
\(61\) 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
\(69\) −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 −0.0848011 − 0.996398i \(-0.527025\pi\)
0.820505 + 0.571639i \(0.193692\pi\)
\(72\) −4.29230 + 8.16066i −0.505852 + 0.961743i
\(73\) −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
\(75\) −7.93290 − 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 0.692542 − 0.721377i \(-0.256491\pi\)
−0.278460 + 0.960448i \(0.589824\pi\)
\(80\) 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 0.216366 − 0.976312i \(-0.430579\pi\)
0.953694 + 0.300778i \(0.0972462\pi\)
\(84\) 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 \(0.157167\pi\)
−0.880559 + 0.473937i \(0.842833\pi\)
\(98\) 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.119.18 yes 104
3.2 odd 2 inner 465.2.t.d.119.36 yes 104
5.4 even 2 inner 465.2.t.d.119.35 yes 104
15.14 odd 2 inner 465.2.t.d.119.17 ✓ 104
31.6 odd 6 inner 465.2.t.d.254.17 yes 104
93.68 even 6 inner 465.2.t.d.254.35 yes 104
155.99 odd 6 inner 465.2.t.d.254.36 yes 104
465.254 even 6 inner 465.2.t.d.254.18 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.17 ✓ 104 15.14 odd 2 inner
465.2.t.d.119.18 yes 104 1.1 even 1 trivial
465.2.t.d.119.35 yes 104 5.4 even 2 inner
465.2.t.d.119.36 yes 104 3.2 odd 2 inner
465.2.t.d.254.17 yes 104 31.6 odd 6 inner
465.2.t.d.254.18 yes 104 465.254 even 6 inner
465.2.t.d.254.35 yes 104 93.68 even 6 inner
465.2.t.d.254.36 yes 104 155.99 odd 6 inner