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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 119.8
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.8

$q$-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-2.04620 q^{2} +(1.68253 - 0.411220i) q^{3} +2.18694 q^{4} +(-1.70071 + 1.45175i) q^{5} +(-3.44279 + 0.841439i) q^{6} +(4.09236 + 2.36273i) q^{7} -0.382512 q^{8} +(2.66180 - 1.38378i) q^{9} +(3.48000 - 2.97057i) q^{10} +(-0.810792 - 1.40433i) q^{11} +(3.67958 - 0.899312i) q^{12} +(-1.28910 - 2.23279i) q^{13} +(-8.37380 - 4.83461i) q^{14} +(-2.26451 + 3.14197i) q^{15} -3.59118 q^{16} +(3.22993 + 1.86480i) q^{17} +(-5.44657 + 2.83149i) q^{18} +(-2.93507 + 5.08370i) q^{19} +(-3.71935 + 3.17488i) q^{20} +(7.85712 + 2.29249i) q^{21} +(1.65904 + 2.87355i) q^{22} +5.26372i q^{23} +(-0.643587 + 0.157297i) q^{24} +(0.784849 - 4.93802i) q^{25} +(2.63776 + 4.56874i) q^{26} +(3.90951 - 3.42283i) q^{27} +(8.94975 + 5.16714i) q^{28} -2.40840 q^{29} +(4.63364 - 6.42911i) q^{30} +(2.69995 + 4.86932i) q^{31} +8.11330 q^{32} +(-1.94167 - 2.02941i) q^{33} +(-6.60909 - 3.81576i) q^{34} +(-10.3900 + 1.92277i) q^{35} +(5.82118 - 3.02624i) q^{36} +(-1.26481 + 2.19071i) q^{37} +(6.00575 - 10.4023i) q^{38} +(-3.08712 - 3.22663i) q^{39} +(0.650543 - 0.555311i) q^{40} +(10.1559 - 5.86354i) q^{41} +(-16.0772 - 4.69090i) q^{42} +(3.70170 - 6.41152i) q^{43} +(-1.77315 - 3.07119i) q^{44} +(-2.51805 + 6.21767i) q^{45} -10.7706i q^{46} +5.01626 q^{47} +(-6.04226 + 1.47676i) q^{48} +(7.66496 + 13.2761i) q^{49} +(-1.60596 + 10.1042i) q^{50} +(6.20129 + 1.80937i) q^{51} +(-2.81919 - 4.88297i) q^{52} +(-0.969781 + 0.559903i) q^{53} +(-7.99964 + 7.00379i) q^{54} +(3.41766 + 1.21130i) q^{55} +(-1.56538 - 0.903771i) q^{56} +(-2.84783 + 9.76042i) q^{57} +4.92806 q^{58} +(-3.27515 - 1.89091i) q^{59} +(-4.95234 + 6.87130i) q^{60} +4.56864i q^{61} +(-5.52463 - 9.96361i) q^{62} +(14.1625 + 0.626178i) q^{63} -9.41908 q^{64} +(5.43385 + 1.92588i) q^{65} +(3.97304 + 4.15259i) q^{66} +(1.65430 - 0.955113i) q^{67} +(7.06366 + 4.07821i) q^{68} +(2.16455 + 8.85636i) q^{69} +(21.2601 - 3.93436i) q^{70} +(-3.93805 + 2.27363i) q^{71} +(-1.01817 + 0.529311i) q^{72} +(6.67236 + 11.5569i) q^{73} +(2.58805 - 4.48263i) q^{74} +(-0.710082 - 8.63109i) q^{75} +(-6.41883 + 11.1177i) q^{76} -7.66272i q^{77} +(6.31686 + 6.60233i) q^{78} +(-13.8533 - 7.99822i) q^{79} +(6.10756 - 5.21349i) q^{80} +(5.17032 - 7.36667i) q^{81} +(-20.7811 + 11.9980i) q^{82} +(-0.966356 + 0.557926i) q^{83} +(17.1830 + 5.01354i) q^{84} +(-8.20041 + 1.51756i) q^{85} +(-7.57441 + 13.1193i) q^{86} +(-4.05219 + 0.990380i) q^{87} +(0.310137 + 0.537174i) q^{88} -7.56756 q^{89} +(5.15244 - 12.7226i) q^{90} -12.1832i q^{91} +11.5114i q^{92} +(6.54510 + 7.08249i) q^{93} -10.2643 q^{94} +(-2.38854 - 12.9069i) q^{95} +(13.6508 - 3.33635i) q^{96} -4.51255i q^{97} +(-15.6841 - 27.1656i) q^{98} +(-4.10145 - 2.61609i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.04620 −1.44688 −0.723441 − 0.690386i \(-0.757441\pi\)
−0.723441 + 0.690386i \(0.757441\pi\)
\(3\) 1.68253 − 0.411220i 0.971408 − 0.237418i
\(4\) 2.18694 1.09347
\(5\) −1.70071 + 1.45175i −0.760582 + 0.649242i
\(6\) −3.44279 + 0.841439i −1.40551 + 0.343516i
\(7\) 4.09236 + 2.36273i 1.54677 + 0.893027i 0.998386 + 0.0567996i \(0.0180896\pi\)
0.548383 + 0.836228i \(0.315244\pi\)
\(8\) −0.382512 −0.135238
\(9\) 2.66180 − 1.38378i 0.887265 − 0.461259i
\(10\) 3.48000 − 2.97057i 1.10047 − 0.939377i
\(11\) −0.810792 − 1.40433i −0.244463 − 0.423422i 0.717518 − 0.696540i \(-0.245278\pi\)
−0.961980 + 0.273118i \(0.911945\pi\)
\(12\) 3.67958 − 0.899312i 1.06220 − 0.259609i
\(13\) −1.28910 − 2.23279i −0.357533 − 0.619265i 0.630015 − 0.776583i \(-0.283049\pi\)
−0.987548 + 0.157318i \(0.949715\pi\)
\(14\) −8.37380 − 4.83461i −2.23799 − 1.29211i
\(15\) −2.26451 + 3.14197i −0.584693 + 0.811254i
\(16\) −3.59118 −0.897795
\(17\) 3.22993 + 1.86480i 0.783374 + 0.452281i 0.837625 − 0.546246i \(-0.183944\pi\)
−0.0542510 + 0.998527i \(0.517277\pi\)
\(18\) −5.44657 + 2.83149i −1.28377 + 0.667388i
\(19\) −2.93507 + 5.08370i −0.673352 + 1.16628i 0.303595 + 0.952801i \(0.401813\pi\)
−0.976948 + 0.213479i \(0.931520\pi\)
\(20\) −3.71935 + 3.17488i −0.831673 + 0.709926i
\(21\) 7.85712 + 2.29249i 1.71456 + 0.500263i
\(22\) 1.65904 + 2.87355i 0.353709 + 0.612642i
\(23\) 5.26372i 1.09756i 0.835966 + 0.548781i \(0.184908\pi\)
−0.835966 + 0.548781i \(0.815092\pi\)
\(24\) −0.643587 + 0.157297i −0.131372 + 0.0321080i
\(25\) 0.784849 − 4.93802i 0.156970 − 0.987603i
\(26\) 2.63776 + 4.56874i 0.517308 + 0.896003i
\(27\) 3.90951 − 3.42283i 0.752385 − 0.658723i
\(28\) 8.94975 + 5.16714i 1.69134 + 0.976497i
\(29\) −2.40840 −0.447228 −0.223614 − 0.974678i \(-0.571785\pi\)
−0.223614 + 0.974678i \(0.571785\pi\)
\(30\) 4.63364 − 6.42911i 0.845983 − 1.17379i
\(31\) 2.69995 + 4.86932i 0.484925 + 0.874556i
\(32\) 8.11330 1.43424
\(33\) −1.94167 − 2.02941i −0.338001 − 0.353276i
\(34\) −6.60909 − 3.81576i −1.13345 − 0.654397i
\(35\) −10.3900 + 1.92277i −1.75623 + 0.325007i
\(36\) 5.82118 − 3.02624i 0.970197 − 0.504373i
\(37\) −1.26481 + 2.19071i −0.207933 + 0.360150i −0.951063 − 0.308996i \(-0.900007\pi\)
0.743130 + 0.669147i \(0.233340\pi\)
\(38\) 6.00575 − 10.4023i 0.974262 − 1.68747i
\(39\) −3.08712 − 3.22663i −0.494335 − 0.516674i
\(40\) 0.650543 − 0.555311i 0.102860 − 0.0878024i
\(41\) 10.1559 − 5.86354i 1.58609 − 0.915731i 0.592150 − 0.805828i \(-0.298279\pi\)
0.993942 − 0.109903i \(-0.0350540\pi\)
\(42\) −16.0772 − 4.69090i −2.48077 − 0.723821i
\(43\) 3.70170 − 6.41152i 0.564503 − 0.977748i −0.432592 − 0.901590i \(-0.642401\pi\)
0.997096 − 0.0761588i \(-0.0242656\pi\)
\(44\) −1.77315 − 3.07119i −0.267313 − 0.462999i
\(45\) −2.51805 + 6.21767i −0.375369 + 0.926875i
\(46\) − 10.7706i − 1.58804i
\(47\) 5.01626 0.731696 0.365848 − 0.930675i \(-0.380779\pi\)
0.365848 + 0.930675i \(0.380779\pi\)
\(48\) −6.04226 + 1.47676i −0.872125 + 0.213153i
\(49\) 7.66496 + 13.2761i 1.09499 + 1.89659i
\(50\) −1.60596 + 10.1042i −0.227117 + 1.42895i
\(51\) 6.20129 + 1.80937i 0.868355 + 0.253362i
\(52\) −2.81919 − 4.88297i −0.390951 − 0.677147i
\(53\) −0.969781 + 0.559903i −0.133210 + 0.0769086i −0.565124 − 0.825006i \(-0.691172\pi\)
0.431914 + 0.901915i \(0.357838\pi\)
\(54\) −7.99964 + 7.00379i −1.08861 + 0.953095i
\(55\) 3.41766 + 1.21130i 0.460838 + 0.163332i
\(56\) −1.56538 − 0.903771i −0.209182 − 0.120772i
\(57\) −2.84783 + 9.76042i −0.377204 + 1.29280i
\(58\) 4.92806 0.647086
\(59\) −3.27515 − 1.89091i −0.426388 − 0.246175i 0.271419 − 0.962461i \(-0.412507\pi\)
−0.697807 + 0.716286i \(0.745841\pi\)
\(60\) −4.95234 + 6.87130i −0.639344 + 0.887081i
\(61\) 4.56864i 0.584954i 0.956272 + 0.292477i \(0.0944796\pi\)
−0.956272 + 0.292477i \(0.905520\pi\)
\(62\) −5.52463 − 9.96361i −0.701629 − 1.26538i
\(63\) 14.1625 + 0.626178i 1.78431 + 0.0788910i
\(64\) −9.41908 −1.17738
\(65\) 5.43385 + 1.92588i 0.673986 + 0.238876i
\(66\) 3.97304 + 4.15259i 0.489048 + 0.511148i
\(67\) 1.65430 − 0.955113i 0.202105 − 0.116686i −0.395532 − 0.918452i \(-0.629440\pi\)
0.597637 + 0.801767i \(0.296106\pi\)
\(68\) 7.06366 + 4.07821i 0.856595 + 0.494555i
\(69\) 2.16455 + 8.85636i 0.260581 + 1.06618i
\(70\) 21.2601 − 3.93436i 2.54107 − 0.470246i
\(71\) −3.93805 + 2.27363i −0.467360 + 0.269830i −0.715134 − 0.698987i \(-0.753634\pi\)
0.247774 + 0.968818i \(0.420301\pi\)
\(72\) −1.01817 + 0.529311i −0.119992 + 0.0623799i
\(73\) 6.67236 + 11.5569i 0.780941 + 1.35263i 0.931394 + 0.364012i \(0.118593\pi\)
−0.150453 + 0.988617i \(0.548073\pi\)
\(74\) 2.58805 − 4.48263i 0.300854 − 0.521095i
\(75\) −0.710082 − 8.63109i −0.0819932 − 0.996633i
\(76\) −6.41883 + 11.1177i −0.736290 + 1.27529i
\(77\) − 7.66272i − 0.873248i
\(78\) 6.31686 + 6.60233i 0.715244 + 0.747566i
\(79\) −13.8533 − 7.99822i −1.55862 − 0.899870i −0.997389 − 0.0722098i \(-0.976995\pi\)
−0.561230 − 0.827660i \(-0.689672\pi\)
\(80\) 6.10756 − 5.21349i 0.682846 − 0.582886i
\(81\) 5.17032 − 7.36667i 0.574480 − 0.818519i
\(82\) −20.7811 + 11.9980i −2.29489 + 1.32495i
\(83\) −0.966356 + 0.557926i −0.106071 + 0.0612403i −0.552097 − 0.833780i \(-0.686172\pi\)
0.446026 + 0.895020i \(0.352839\pi\)
\(84\) 17.1830 + 5.01354i 1.87482 + 0.547022i
\(85\) −8.20041 + 1.51756i −0.889460 + 0.164602i
\(86\) −7.57441 + 13.1193i −0.816770 + 1.41469i
\(87\) −4.05219 + 0.990380i −0.434440 + 0.106180i
\(88\) 0.310137 + 0.537174i 0.0330608 + 0.0572629i
\(89\) −7.56756 −0.802160 −0.401080 − 0.916043i \(-0.631365\pi\)
−0.401080 + 0.916043i \(0.631365\pi\)
\(90\) 5.15244 − 12.7226i 0.543115 − 1.34108i
\(91\) − 12.1832i − 1.27715i
\(92\) 11.5114i 1.20015i
\(93\) 6.54510 + 7.08249i 0.678695 + 0.734420i
\(94\) −10.2643 −1.05868
\(95\) −2.38854 − 12.9069i −0.245059 − 1.32422i
\(96\) 13.6508 − 3.33635i 1.39323 − 0.340515i
\(97\) − 4.51255i − 0.458180i −0.973405 − 0.229090i \(-0.926425\pi\)
0.973405 − 0.229090i \(-0.0735749\pi\)
\(98\) −15.6841 − 27.1656i −1.58433 − 2.74414i
\(99\) −4.10145 − 2.61609i −0.412211 − 0.262927i
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.119.8 yes 104
3.2 odd 2 inner 465.2.t.d.119.46 yes 104
5.4 even 2 inner 465.2.t.d.119.45 yes 104
15.14 odd 2 inner 465.2.t.d.119.7 ✓ 104
31.6 odd 6 inner 465.2.t.d.254.7 yes 104
93.68 even 6 inner 465.2.t.d.254.45 yes 104
155.99 odd 6 inner 465.2.t.d.254.46 yes 104
465.254 even 6 inner 465.2.t.d.254.8 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.7 ✓ 104 15.14 odd 2 inner
465.2.t.d.119.8 yes 104 1.1 even 1 trivial
465.2.t.d.119.45 yes 104 5.4 even 2 inner
465.2.t.d.119.46 yes 104 3.2 odd 2 inner
465.2.t.d.254.7 yes 104 31.6 odd 6 inner
465.2.t.d.254.8 yes 104 465.254 even 6 inner
465.2.t.d.254.45 yes 104 93.68 even 6 inner
465.2.t.d.254.46 yes 104 155.99 odd 6 inner