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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 119.35
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.35

$q$-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} +(-1.65152 + 1.50748i) q^{5} +(0.997522 + 1.65176i) q^{6} +(-1.74935 - 1.00999i) q^{7} -3.07355 q^{8} +(-1.39653 + 2.65513i) q^{9} +(-1.83989 + 1.67941i) 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} +(-3.71383 - 1.09885i) q^{15} -1.90634 q^{16} +(-0.447757 - 0.258512i) q^{17} +(-1.55581 + 2.95796i) q^{18} +(-1.42743 + 2.47239i) q^{19} +(1.25330 - 1.14399i) q^{20} +(-0.0688986 - 3.49802i) q^{21} +(0.802452 + 1.38989i) q^{22} +7.25758i q^{23} +(-2.75204 - 4.55701i) 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} +(-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} +(1.05980 - 2.01492i) 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} +(-0.0767569 - 3.89699i) q^{42} +(1.20627 - 2.08932i) q^{43} +(-0.546619 - 0.946772i) q^{44} +(-1.69614 - 6.49023i) q^{45} +8.08535i q^{46} +8.36312 q^{47} +(-1.70693 - 2.82645i) q^{48} +(-1.45985 - 2.52854i) q^{49} +(0.506934 - 5.54716i) 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} +(-3.07030 - 0.974592i) q^{55} +(5.37670 + 3.10424i) q^{56} +(-4.94382 + 0.0973758i) q^{57} +7.75474 q^{58} +(10.0708 + 5.81440i) q^{59} +(2.81835 + 0.833891i) q^{60} +9.17896i q^{61} +(-2.46449 - 5.69219i) q^{62} +(5.12466 - 3.23426i) q^{63} +8.29489 q^{64} +(-4.23827 - 1.34534i) 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} +(7.78995 - 3.78374i) 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} +(-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} +(-1.88960 - 7.23048i) 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 \(-0.371133\pi\)
0.393878 + 0.919163i \(0.371133\pi\)
\(3\) 0.895396 + 1.48265i 0.516957 + 0.856011i
\(4\) −0.758880 −0.379440
\(5\) −1.65152 + 1.50748i −0.738582 + 0.674164i
\(6\) 0.997522 + 1.65176i 0.407236 + 0.674328i
\(7\) −1.74935 − 1.00999i −0.661191 − 0.381739i 0.131539 − 0.991311i \(-0.458008\pi\)
−0.792731 + 0.609572i \(0.791341\pi\)
\(8\) −3.07355 −1.08666
\(9\) −1.39653 + 2.65513i −0.465510 + 0.885043i
\(10\) −1.83989 + 1.67941i −0.581823 + 0.531077i
\(11\) 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 0.970329 − 0.241787i \(-0.0777336\pi\)
−0.694559 + 0.719436i \(0.744400\pi\)
\(14\) −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 \(0.686637\pi\)
\(18\) −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 \(0.272865\pi\)
\(20\) 1.25330 − 1.14399i 0.280247 − 0.255804i
\(21\) −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
\(25\) 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 \(-0.276317\pi\)
0.646296 + 0.763087i \(0.276317\pi\)
\(30\) −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
\(36\) 1.05980 − 2.01492i 0.176633 − 0.335820i
\(37\) −3.15397 + 5.46283i −0.518509 + 0.898084i 0.481260 + 0.876578i \(0.340179\pi\)
−0.999769 + 0.0215061i \(0.993154\pi\)
\(38\) −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 \(-0.501282\pi\)
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 −0.759269 − 0.650777i \(-0.774443\pi\)
0.943224 + 0.332158i \(0.107777\pi\)
\(44\) −0.546619 − 0.946772i −0.0824059 − 0.142731i
\(45\) −1.69614 − 6.49023i −0.252846 − 0.967507i
\(46\) 8.08535i 1.19212i
\(47\) 8.36312 1.21989 0.609944 − 0.792445i \(-0.291192\pi\)
0.609944 + 0.792445i \(0.291192\pi\)
\(48\) −1.70693 − 2.82645i −0.246374 − 0.407963i
\(49\) −1.45985 − 2.52854i −0.208551 − 0.361220i
\(50\) 0.506934 − 5.54716i 0.0716914 − 0.784487i
\(51\) −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 \(-0.875187\pi\)
−0.131108 + 0.991368i \(0.541853\pi\)
\(54\) −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
\(57\) −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 \(0.393345\pi\)
\(60\) 2.81835 + 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
\(65\) −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 −0.667275 − 0.744812i \(-0.732539\pi\)
0.311389 + 0.950283i \(0.399206\pi\)
\(68\) 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 0.972779 − 0.231733i \(-0.0744396\pi\)
−0.687076 + 0.726585i \(0.741106\pi\)
\(74\) −3.51370 + 6.08590i −0.408459 + 0.707472i
\(75\) 7.78995 − 3.78374i 0.899506 − 0.436909i
\(76\) 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 \(0.589824\pi\)
\(80\) 3.14836 − 2.87376i 0.351998 − 0.321297i
\(81\) −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 \(-0.902754\pi\)
−0.216366 + 0.976312i \(0.569421\pi\)
\(84\) 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 \(0.438156\pi\)
\(90\) −1.88960 − 7.23048i −0.199181 − 0.762160i
\(91\) − 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.119.35 yes 104
3.2 odd 2 inner 465.2.t.d.119.17 ✓ 104
5.4 even 2 inner 465.2.t.d.119.18 yes 104
15.14 odd 2 inner 465.2.t.d.119.36 yes 104
31.6 odd 6 inner 465.2.t.d.254.36 yes 104
93.68 even 6 inner 465.2.t.d.254.18 yes 104
155.99 odd 6 inner 465.2.t.d.254.17 yes 104
465.254 even 6 inner 465.2.t.d.254.35 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.17 ✓ 104 3.2 odd 2 inner
465.2.t.d.119.18 yes 104 5.4 even 2 inner
465.2.t.d.119.35 yes 104 1.1 even 1 trivial
465.2.t.d.119.36 yes 104 15.14 odd 2 inner
465.2.t.d.254.17 yes 104 155.99 odd 6 inner
465.2.t.d.254.18 yes 104 93.68 even 6 inner
465.2.t.d.254.35 yes 104 465.254 even 6 inner
465.2.t.d.254.36 yes 104 31.6 odd 6 inner