LMFDB - Embedded newform 162.13.b.c.161.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,13,Mod(161,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.161"); S:= CuspForms(chi, 13); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 13, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 13 $
Character orbit:	$[\chi]$	$=$	162.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,-49152,0,0,137280] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$148.066998399$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.15
Character	$\chi$	$=$	162.161
Dual form			162.13.b.c.161.10

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+45.2548i q^{2} -2048.00 q^{4} -17296.5i q^{5} +10199.1 q^{7} -92681.9i q^{8} +782751. q^{10} +1.12395e6i q^{11} +6.90443e6 q^{13} +461558. i q^{14} +4.19430e6 q^{16} -1.63973e7i q^{17} +5.55128e7 q^{19} +3.54233e7i q^{20} -5.08642e7 q^{22} -1.31372e8i q^{23} -5.50292e7 q^{25} +3.12459e8i q^{26} -2.08877e7 q^{28} +6.55877e8i q^{29} -1.26703e8 q^{31} +1.89813e8i q^{32} +7.42055e8 q^{34} -1.76409e8i q^{35} -4.30043e9 q^{37} +2.51222e9i q^{38} -1.60307e9 q^{40} +7.66124e9i q^{41} +6.29795e9 q^{43} -2.30185e9i q^{44} +5.94521e9 q^{46} -1.24592e10i q^{47} -1.37373e10 q^{49} -2.49034e9i q^{50} -1.41403e10 q^{52} +1.58497e10i q^{53} +1.94404e10 q^{55} -9.45271e8i q^{56} -2.96816e10 q^{58} +3.86375e10i q^{59} -4.54381e9 q^{61} -5.73393e9i q^{62} -8.58993e9 q^{64} -1.19423e11i q^{65} -7.99165e10 q^{67} +3.35816e10i q^{68} +7.98335e9 q^{70} -8.04417e10i q^{71} +2.85166e11 q^{73} -1.94615e11i q^{74} -1.13690e11 q^{76} +1.14633e10i q^{77} +2.92820e11 q^{79} -7.25469e10i q^{80} -3.46708e11 q^{82} -3.83034e11i q^{83} -2.83616e11 q^{85} +2.85013e11i q^{86} +1.04170e11 q^{88} -5.03577e11i q^{89} +7.04189e10 q^{91} +2.69050e11i q^{92} +5.63841e11 q^{94} -9.60178e11i q^{95} +6.80045e11 q^{97} -6.21678e11i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 49152 q^{4} + 137280 q^{7} - 1700160 q^{13} + 100663296 q^{16} - 47432760 q^{19} + 134668800 q^{22} - 1721170488 q^{25} - 281149440 q^{28} - 2023332000 q^{31} + 755573760 q^{34} - 8737330560 q^{37}+ \cdots + 1991581150440 q^{97}+O(q^{100}) $ 24 * q - 49152 * q^4 + 137280 * q^7 - 1700160 * q^13 + 100663296 * q^16 - 47432760 * q^19 + 134668800 * q^22 - 1721170488 * q^25 - 281149440 * q^28 - 2023332000 * q^31 + 755573760 * q^34 - 8737330560 * q^37 + 2564630040 * q^43 + 16593325056 * q^46 + 76678938504 * q^49 + 3481927680 * q^52 - 82896170256 * q^55 - 2930457600 * q^58 + 144178091904 * q^61 - 206158430208 * q^64 + 364748716920 * q^67 - 150468074496 * q^70 - 404658904200 * q^73 + 97142292480 * q^76 - 255432049584 * q^79 - 966809364480 * q^82 + 341523808704 * q^85 - 275801702400 * q^88 - 563012378112 * q^91 - 311763336192 * q^94 + 1991581150440 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	45.2548i	0.707107i
$2$	45.2548i	0.707107i
$3$	0	0
$3$	0	0
$4$	−2048.00	−0.500000
$5$	− 17296.5i	− 1.10698i	−0.832857	−	0.553489i	$-0.813296\pi$
$5$	− 17296.5i	− 1.10698i	0.832857	−	0.553489i	$-0.186704\pi$
$6$	0	0
$7$	10199.1	0.0866908	0.0433454	−	0.999060i	$-0.486198\pi$
$7$	10199.1	0.0866908	0.0433454	+	0.999060i	$0.486198\pi$
$8$	− 92681.9i	− 0.353553i
$9$	0	0
$10$	782751.	0.782751
$11$	1.12395e6i	0.634440i	0.948352	+	0.317220i	$0.102749\pi$
$11$	1.12395e6i	0.634440i	−0.948352	+	0.317220i	$0.897251\pi$
$12$	0	0
$13$	6.90443e6	1.43043	0.715216	−	0.698903i	$-0.246328\pi$
$13$	6.90443e6	1.43043	0.715216	+	0.698903i	$0.246328\pi$
$14$	461558.i	0.0612997i
$15$	0	0
$16$	4.19430e6	0.250000
$17$	− 1.63973e7i	− 0.679325i	−0.940547	−	0.339663i	$-0.889687\pi$
$17$	− 1.63973e7i	− 0.679325i	0.940547	−	0.339663i	$-0.110313\pi$
$18$	0	0
$19$	5.55128e7	1.17997	0.589986	−	0.807414i	$-0.299133\pi$
$19$	5.55128e7	1.17997	0.589986	+	0.807414i	$0.299133\pi$
$20$	3.54233e7i	0.553489i
$21$	0	0
$22$	−5.08642e7	−0.448617
$23$	− 1.31372e8i	− 0.887433i	−0.896167	−	0.443716i	$-0.853660\pi$
$23$	− 1.31372e8i	− 0.887433i	0.896167	−	0.443716i	$-0.146340\pi$
$24$	0	0
$25$	−5.50292e7	−0.225400
$26$	3.12459e8i	1.01147i
$27$	0	0
$28$	−2.08877e7	−0.0433454
$29$	6.55877e8i	1.10264i	0.834293	+	0.551321i	$0.185876\pi$
$29$	6.55877e8i	1.10264i	−0.834293	+	0.551321i	$0.814124\pi$
$30$	0	0
$31$	−1.26703e8	−0.142764	−0.0713818	−	0.997449i	$-0.522741\pi$
$31$	−1.26703e8	−0.142764	−0.0713818	+	0.997449i	$0.522741\pi$
$32$	1.89813e8i	0.176777i
$33$	0	0
$34$	7.42055e8	0.480356
$35$	− 1.76409e8i	− 0.0959648i
$36$	0	0
$37$	−4.30043e9	−1.67611	−0.838053	−	0.545588i	$-0.816306\pi$
$37$	−4.30043e9	−1.67611	−0.838053	+	0.545588i	$0.816306\pi$
$38$	2.51222e9i	0.834366i
$39$	0	0
$40$	−1.60307e9	−0.391376
$41$	7.66124e9i	1.61286i	0.591332	+	0.806428i	$0.298602\pi$
$41$	7.66124e9i	1.61286i	−0.591332	+	0.806428i	$0.701398\pi$
$42$	0	0
$43$	6.29795e9	0.996296	0.498148	−	0.867092i	$-0.334014\pi$
$43$	6.29795e9	0.996296	0.498148	+	0.867092i	$0.334014\pi$
$44$	− 2.30185e9i	− 0.317220i
$45$	0	0
$46$	5.94521e9	0.627510
$47$	− 1.24592e10i	− 1.15586i	−0.816087	−	0.577929i	$-0.803861\pi$
$47$	− 1.24592e10i	− 1.15586i	0.816087	−	0.577929i	$-0.196139\pi$
$48$	0	0
$49$	−1.37373e10	−0.992485
$50$	− 2.49034e9i	− 0.159382i
$51$	0	0
$52$	−1.41403e10	−0.715216
$53$	1.58497e10i	0.715097i	0.933895	+	0.357548i	$0.116387\pi$
$53$	1.58497e10i	0.715097i	−0.933895	+	0.357548i	$0.883613\pi$
$54$	0	0
$55$	1.94404e10	0.702311
$56$	− 9.45271e8i	− 0.0306498i
$57$	0	0
$58$	−2.96816e10	−0.779686
$59$	3.86375e10i	0.916004i	0.888951	+	0.458002i	$0.151435\pi$
$59$	3.86375e10i	0.916004i	−0.888951	+	0.458002i	$0.848565\pi$
$60$	0	0
$61$	−4.54381e9	−0.0881944	−0.0440972	−	0.999027i	$-0.514041\pi$
$61$	−4.54381e9	−0.0881944	−0.0440972	+	0.999027i	$0.514041\pi$
$62$	− 5.73393e9i	− 0.100949i
$63$	0	0
$64$	−8.58993e9	−0.125000
$65$	− 1.19423e11i	− 1.58346i
$66$	0	0
$67$	−7.99165e10	−0.883462	−0.441731	−	0.897148i	$-0.645635\pi$
$67$	−7.99165e10	−0.883462	−0.441731	+	0.897148i	$0.645635\pi$
$68$	3.35816e10i	0.339663i
$69$	0	0
$70$	7.98335e9	0.0678574
$71$	− 8.04417e10i	− 0.627959i	−0.949430	−	0.313979i	$-0.898338\pi$
$71$	− 8.04417e10i	− 0.627959i	0.949430	−	0.313979i	$-0.101662\pi$
$72$	0	0
$73$	2.85166e11	1.88434	0.942172	−	0.335130i	$-0.108780\pi$
$73$	2.85166e11	1.88434	0.942172	+	0.335130i	$0.108780\pi$
$74$	− 1.94615e11i	− 1.18519i
$75$	0	0
$76$	−1.13690e11	−0.589986
$77$	1.14633e10i	0.0550002i
$78$	0	0
$79$	2.92820e11	1.20459	0.602294	−	0.798274i	$-0.294253\pi$
$79$	2.92820e11	1.20459	0.602294	+	0.798274i	$0.294253\pi$
$80$	− 7.25469e10i	− 0.276744i
$81$	0	0
$82$	−3.46708e11	−1.14046
$83$	− 3.83034e11i	− 1.17157i	−0.810466	−	0.585786i	$-0.800786\pi$
$83$	− 3.83034e11i	− 1.17157i	0.810466	−	0.585786i	$-0.199214\pi$
$84$	0	0
$85$	−2.83616e11	−0.751998
$86$	2.85013e11i	0.704488i
$87$	0	0
$88$	1.04170e11	0.224309
$89$	− 5.03577e11i	− 1.01327i	−0.862160	−	0.506635i	$-0.830889\pi$
$89$	− 5.03577e11i	− 1.01327i	0.862160	−	0.506635i	$-0.169111\pi$
$90$	0	0
$91$	7.04189e10	0.124005
$92$	2.69050e11i	0.443716i
$93$	0	0
$94$	5.63841e11	0.817315
$95$	− 9.60178e11i	− 1.30620i
$96$	0	0
$97$	6.80045e11	0.816408	0.408204	−	0.912891i	$-0.366155\pi$
$97$	6.80045e11	0.816408	0.408204	+	0.912891i	$0.366155\pi$
$98$	− 6.21678e11i	− 0.701793i
$99$	0	0
$100$	1.12700e11	0.112700
$101$	2.35393e11i	0.221751i	0.993834	+	0.110875i	$0.0353654\pi$
$101$	2.35393e11i	0.221751i	−0.993834	+	0.110875i	$0.964635\pi$
$102$	0	0
$103$	6.99404e11	0.585739	0.292870	−	0.956152i	$-0.405390\pi$
$103$	6.99404e11	0.585739	0.292870	+	0.956152i	$0.405390\pi$
$104$	− 6.39915e11i	− 0.505734i
$105$	0	0
$106$	−7.17274e11	−0.505650
$107$	2.83169e12i	1.88687i	0.331551	+	0.943437i	$0.392428\pi$
$107$	2.83169e12i	1.88687i	−0.331551	+	0.943437i	$0.607572\pi$
$108$	0	0
$109$	2.14470e12	1.27882	0.639408	−	0.768867i	$-0.279179\pi$
$109$	2.14470e12	1.27882	0.639408	+	0.768867i	$0.279179\pi$
$110$	8.79773e11i	0.496609i
$111$	0	0
$112$	4.27781e10	0.0216727
$113$	1.86036e10i	0.00893565i	0.999990	+	0.00446782i	$0.00142216\pi$
$113$	1.86036e10i	0.00893565i	−0.999990	+	0.00446782i	$0.998578\pi$
$114$	0	0
$115$	−2.27228e12	−0.982368
$116$	− 1.34324e12i	− 0.551321i
$117$	0	0
$118$	−1.74853e12	−0.647713
$119$	− 1.67237e11i	− 0.0588913i
$120$	0	0
$121$	1.87517e12	0.597486
$122$	− 2.05629e11i	− 0.0623628i
$123$	0	0
$124$	2.59488e11	0.0713818
$125$	− 3.27097e12i	− 0.857465i
$126$	0	0
$127$	−4.18875e12	−0.998302	−0.499151	−	0.866515i	$-0.666355\pi$
$127$	−4.18875e12	−0.998302	−0.499151	+	0.866515i	$0.666355\pi$
$128$	− 3.88736e11i	− 0.0883883i
$129$	0	0
$130$	5.40445e12	1.11967
$131$	− 5.03918e12i	− 0.997084i	−0.866865	−	0.498542i	$-0.833869\pi$
$131$	− 5.03918e12i	− 0.997084i	0.866865	−	0.498542i	$-0.166131\pi$
$132$	0	0
$133$	5.66180e11	0.102293
$134$	− 3.61661e12i	− 0.624702i
$135$	0	0
$136$	−1.51973e12	−0.240178
$137$	− 5.66300e12i	− 0.856491i	−0.903662	−	0.428245i	$-0.859132\pi$
$137$	− 5.66300e12i	− 0.856491i	0.903662	−	0.428245i	$-0.140868\pi$
$138$	0	0
$139$	2.56146e12	0.355139	0.177569	−	0.984108i	$-0.443177\pi$
$139$	2.56146e12	0.355139	0.177569	+	0.984108i	$0.443177\pi$
$140$	3.61285e11i	0.0479824i
$141$	0	0
$142$	3.64038e12	0.444034
$143$	7.76023e12i	0.907524i
$144$	0	0
$145$	1.13444e13	1.22060
$146$	1.29051e13i	1.33243i
$147$	0	0
$148$	8.80728e12	0.838053
$149$	− 2.03049e13i	− 1.85560i	−0.373080	−	0.927799i	$-0.621698\pi$
$149$	− 2.03049e13i	− 1.85560i	0.373080	−	0.927799i	$-0.378302\pi$
$150$	0	0
$151$	2.68907e12	0.226851	0.113425	−	0.993547i	$-0.463818\pi$
$151$	2.68907e12	0.226851	0.113425	+	0.993547i	$0.463818\pi$
$152$	− 5.14503e12i	− 0.417183i
$153$	0	0
$154$	−5.18768e11	−0.0388910
$155$	2.19153e12i	0.158036i
$156$	0	0
$157$	−2.09254e13	−1.39726	−0.698628	−	0.715486i	$-0.746206\pi$
$157$	−2.09254e13	−1.39726	−0.698628	+	0.715486i	$0.746206\pi$
$158$	1.32515e13i	0.851773i
$159$	0	0
$160$	3.28310e12	0.195688
$161$	− 1.33987e12i	− 0.0769323i
$162$	0	0
$163$	1.89124e13	1.00837	0.504186	−	0.863595i	$-0.331793\pi$
$163$	1.89124e13	1.00837	0.504186	+	0.863595i	$0.331793\pi$
$164$	− 1.56902e13i	− 0.806428i
$165$	0	0
$166$	1.73341e13	0.828426
$167$	2.82901e13i	1.30418i	0.758143	+	0.652088i	$0.226107\pi$
$167$	2.82901e13i	1.30418i	−0.758143	+	0.652088i	$0.773893\pi$
$168$	0	0
$169$	2.43730e13	1.04614
$170$	− 1.28350e13i	− 0.531743i
$171$	0	0
$172$	−1.28982e13	−0.498148
$173$	1.23231e13i	0.459667i	0.973230	+	0.229833i	$0.0738182\pi$
$173$	1.23231e13i	0.459667i	−0.973230	+	0.229833i	$0.926182\pi$
$174$	0	0
$175$	−5.61248e11	−0.0195401
$176$	4.71419e12i	0.158610i
$177$	0	0
$178$	2.27893e13	0.716491
$179$	4.25445e13i	1.29338i	0.762754	+	0.646689i	$0.223847\pi$
$179$	4.25445e13i	1.29338i	−0.762754	+	0.646689i	$0.776153\pi$
$180$	0	0
$181$	3.90210e13	1.10975	0.554877	−	0.831932i	$-0.312765\pi$
$181$	3.90210e13	1.10975	0.554877	+	0.831932i	$0.312765\pi$
$182$	3.18679e12i	0.0876851i
$183$	0	0
$184$	−1.21758e13	−0.313755
$185$	7.43825e13i	1.85541i
$186$	0	0
$187$	1.84297e13	0.430991
$188$	2.55165e13i	0.577929i
$189$	0	0
$190$	4.34527e13	0.923624
$191$	− 6.78891e13i	− 1.39830i	−0.714976	−	0.699149i	$-0.753562\pi$
$191$	− 6.78891e13i	− 1.39830i	0.714976	−	0.699149i	$-0.246438\pi$
$192$	0	0
$193$	3.94023e13	0.762391	0.381196	−	0.924494i	$-0.375512\pi$
$193$	3.94023e13	0.762391	0.381196	+	0.924494i	$0.375512\pi$
$194$	3.07753e13i	0.577288i
$195$	0	0
$196$	2.81339e13	0.496242
$197$	2.19958e13i	0.376307i	0.982140	+	0.188153i	$0.0602502\pi$
$197$	2.19958e13i	0.376307i	−0.982140	+	0.188153i	$0.939750\pi$
$198$	0	0
$199$	8.52869e13	1.37330	0.686648	−	0.726990i	$-0.259081\pi$
$199$	8.52869e13	1.37330	0.686648	+	0.726990i	$0.259081\pi$
$200$	5.10021e12i	0.0796908i
$201$	0	0
$202$	−1.06527e13	−0.156801
$203$	6.68935e12i	0.0955889i
$204$	0	0
$205$	1.32513e14	1.78540
$206$	3.16514e13i	0.414180i
$207$	0	0
$208$	2.89593e13	0.357608
$209$	6.23936e13i	0.748621i
$210$	0	0
$211$	1.21056e14	1.37180	0.685900	−	0.727696i	$-0.259409\pi$
$211$	1.21056e14	1.37180	0.685900	+	0.727696i	$0.259409\pi$
$212$	− 3.24601e13i	− 0.357548i
$213$	0	0
$214$	−1.28148e14	−1.33422
$215$	− 1.08933e14i	− 1.10288i
$216$	0	0
$217$	−1.29226e12	−0.0123763
$218$	9.70582e13i	0.904260i
$219$	0	0
$220$	−3.98140e13	−0.351156
$221$	− 1.13214e14i	− 0.971729i
$222$	0	0
$223$	−6.40751e13	−0.521027	−0.260513	−	0.965470i	$-0.583892\pi$
$223$	−6.40751e13	−0.521027	−0.260513	+	0.965470i	$0.583892\pi$
$224$	1.93592e12i	0.0153249i
$225$	0	0
$226$	−8.41902e11	−0.00631846
$227$	− 1.16817e13i	− 0.0853790i	−0.999088	−	0.0426895i	$-0.986407\pi$
$227$	− 1.16817e13i	− 0.0853790i	0.999088	−	0.0426895i	$-0.0135926\pi$
$228$	0	0
$229$	1.35025e14	0.936269	0.468134	−	0.883657i	$-0.344926\pi$
$229$	1.35025e14	0.936269	0.468134	+	0.883657i	$0.344926\pi$
$230$	− 1.02832e14i	− 0.694639i
$231$	0	0
$232$	6.07879e13	0.389843
$233$	− 7.15783e13i	− 0.447348i	−0.974664	−	0.223674i	$-0.928195\pi$
$233$	− 7.15783e13i	− 0.447348i	0.974664	−	0.223674i	$-0.0718052\pi$
$234$	0	0
$235$	−2.15502e14	−1.27951
$236$	− 7.91297e13i	− 0.458002i
$237$	0	0
$238$	7.56829e12	0.0416424
$239$	− 2.12746e14i	− 1.14150i	−0.821125	−	0.570748i	$-0.806653\pi$
$239$	− 2.12746e14i	− 1.14150i	0.821125	−	0.570748i	$-0.193347\pi$
$240$	0	0
$241$	2.18197e14	1.11364	0.556821	−	0.830632i	$-0.312021\pi$
$241$	2.18197e14	1.11364	0.556821	+	0.830632i	$0.312021\pi$
$242$	8.48603e13i	0.422486i
$243$	0	0
$244$	9.30572e12	0.0440972
$245$	2.37607e14i	1.09866i
$246$	0	0
$247$	3.83284e14	1.68787
$248$	1.17431e13i	0.0504746i
$249$	0	0
$250$	1.48027e14	0.606320
$251$	− 2.22583e14i	− 0.890123i	−0.895500	−	0.445062i	$-0.853182\pi$
$251$	− 2.22583e14i	− 0.890123i	0.895500	−	0.445062i	$-0.146818\pi$
$252$	0	0
$253$	1.47655e14	0.563023
$254$	− 1.89561e14i	− 0.705906i
$255$	0	0
$256$	1.75922e13	0.0625000
$257$	8.34673e13i	0.289679i	0.989455	+	0.144840i	$0.0462666\pi$
$257$	8.34673e13i	0.289679i	−0.989455	+	0.144840i	$0.953733\pi$
$258$	0	0
$259$	−4.38605e13	−0.145303
$260$	2.44577e14i	0.791729i
$261$	0	0
$262$	2.28047e14	0.705045
$263$	1.28364e14i	0.387891i	0.981012	+	0.193946i	$0.0621285\pi$
$263$	1.28364e14i	0.387891i	−0.981012	+	0.193946i	$0.937871\pi$
$264$	0	0
$265$	2.74144e14	0.791596
$266$	2.56224e13i	0.0723319i
$267$	0	0
$268$	1.63669e14	0.441731
$269$	− 3.01491e14i	− 0.795721i	−0.917446	−	0.397861i	$-0.869753\pi$
$269$	− 3.01491e14i	− 0.795721i	0.917446	−	0.397861i	$-0.130247\pi$
$270$	0	0
$271$	4.73030e13	0.119419	0.0597094	−	0.998216i	$-0.480983\pi$
$271$	4.73030e13	0.119419	0.0597094	+	0.998216i	$0.480983\pi$
$272$	− 6.87751e13i	− 0.169831i
$273$	0	0
$274$	2.56278e14	0.605631
$275$	− 6.18500e13i	− 0.143003i
$276$	0	0
$277$	−6.36225e14	−1.40842	−0.704209	−	0.709992i	$-0.748698\pi$
$277$	−6.36225e14	−1.40842	−0.704209	+	0.709992i	$0.748698\pi$
$278$	1.15918e14i	0.251121i
$279$	0	0
$280$	−1.63499e13	−0.0339287
$281$	− 4.65244e14i	− 0.945023i	−0.881324	−	0.472512i	$-0.843347\pi$
$281$	− 4.65244e14i	− 0.945023i	0.881324	−	0.472512i	$-0.156653\pi$
$282$	0	0
$283$	−7.23115e14	−1.40763	−0.703815	−	0.710383i	$-0.748522\pi$
$283$	−7.23115e14	−1.40763	−0.703815	+	0.710383i	$0.748522\pi$
$284$	1.64745e14i	0.313979i
$285$	0	0
$286$	−3.51188e14	−0.641716
$287$	7.81377e13i	0.139820i
$288$	0	0
$289$	3.13752e14	0.538517
$290$	5.13389e14i	0.863094i
$291$	0	0
$292$	−5.84019e14	−0.942172
$293$	− 2.57869e14i	− 0.407562i	−0.979017	−	0.203781i	$-0.934677\pi$
$293$	− 2.57869e14i	− 0.407562i	0.979017	−	0.203781i	$-0.0653230\pi$
$294$	0	0
$295$	6.68295e14	1.01400
$296$	3.98572e14i	0.592593i
$297$	0	0
$298$	9.18897e14	1.31211
$299$	− 9.07048e14i	− 1.26941i
$300$	0	0
$301$	6.42333e13	0.0863697
$302$	1.21693e14i	0.160408i
$303$	0	0
$304$	2.32837e14	0.294993
$305$	7.85921e13i	0.0976292i
$306$	0	0
$307$	−6.19760e14	−0.740275	−0.370138	−	0.928977i	$-0.620689\pi$
$307$	−6.19760e14	−0.740275	−0.370138	+	0.928977i	$0.620689\pi$
$308$	− 2.34768e13i	− 0.0275001i
$309$	0	0
$310$	−9.91771e13	−0.111748
$311$	8.81568e14i	0.974302i	0.873318	+	0.487151i	$0.161964\pi$
$311$	8.81568e14i	0.974302i	−0.873318	+	0.487151i	$0.838036\pi$
$312$	0	0
$313$	1.41252e15	1.50220	0.751100	−	0.660189i	$-0.229524\pi$
$313$	1.41252e15	1.50220	0.751100	+	0.660189i	$0.229524\pi$
$314$	− 9.46975e14i	− 0.988009i
$315$	0	0
$316$	−5.99696e14	−0.602294
$317$	− 1.36168e15i	− 1.34190i	−0.741501	−	0.670951i	$-0.765886\pi$
$317$	− 1.36168e15i	− 1.34190i	0.741501	−	0.670951i	$-0.234114\pi$
$318$	0	0
$319$	−7.37173e14	−0.699560
$320$	1.48576e14i	0.138372i
$321$	0	0
$322$	6.06358e13	0.0543993
$323$	− 9.10258e14i	− 0.801584i
$324$	0	0
$325$	−3.79945e14	−0.322419
$326$	8.55877e14i	0.713027i
$327$	0	0
$328$	7.10058e14	0.570231
$329$	− 1.27073e14i	− 0.100202i
$330$	0	0
$331$	−1.90273e15	−1.44680	−0.723401	−	0.690428i	$-0.757422\pi$
$331$	−1.90273e15	−1.44680	−0.723401	+	0.690428i	$0.757422\pi$
$332$	7.84454e14i	0.585786i
$333$	0	0
$334$	−1.28027e15	−0.922192
$335$	1.38228e15i	0.977973i
$336$	0	0
$337$	−2.16627e15	−1.47888	−0.739442	−	0.673220i	$-0.764910\pi$
$337$	−2.16627e15	−1.47888	−0.739442	+	0.673220i	$0.764910\pi$
$338$	1.10300e15i	0.739731i
$339$	0	0
$340$	5.80845e14	0.375999
$341$	− 1.42408e14i	− 0.0905750i
$342$	0	0
$343$	−2.81276e14	−0.172730
$344$	− 5.83706e14i	− 0.352244i
$345$	0	0
$346$	−5.57680e14	−0.325034
$347$	− 6.41400e14i	− 0.367411i	−0.982981	−	0.183705i	$-0.941191\pi$
$347$	− 6.41400e14i	− 0.367411i	0.982981	−	0.183705i	$-0.0588093\pi$
$348$	0	0
$349$	−1.21045e15	−0.669877	−0.334939	−	0.942240i	$-0.608716\pi$
$349$	−1.21045e15	−0.669877	−0.334939	+	0.942240i	$0.608716\pi$
$350$	− 2.53992e13i	− 0.0138169i
$351$	0	0
$352$	−2.13340e14	−0.112154
$353$	− 8.68674e14i	− 0.448961i	−0.974479	−	0.224481i	$-0.927931\pi$
$353$	− 8.68674e14i	− 0.448961i	0.974479	−	0.224481i	$-0.0720685\pi$
$354$	0	0
$355$	−1.39136e15	−0.695136
$356$	1.03133e15i	0.506635i
$357$	0	0
$358$	−1.92535e15	−0.914557
$359$	1.63259e15i	0.762623i	0.924447	+	0.381312i	$0.124527\pi$
$359$	1.63259e15i	0.762623i	−0.924447	+	0.381312i	$0.875473\pi$
$360$	0	0
$361$	8.68354e14	0.392332
$362$	1.76589e15i	0.784714i
$363$	0	0
$364$	−1.44218e14	−0.0620027
$365$	− 4.93238e15i	− 2.08593i
$366$	0	0
$367$	−3.43692e15	−1.40661	−0.703304	−	0.710889i	$-0.748293\pi$
$367$	−3.43692e15	−1.40661	−0.703304	+	0.710889i	$0.748293\pi$
$368$	− 5.51014e14i	− 0.221858i
$369$	0	0
$370$	−3.36617e15	−1.31197
$371$	1.61652e14i	0.0619923i
$372$	0	0
$373$	2.25815e15	0.838493	0.419247	−	0.907872i	$-0.362294\pi$
$373$	2.25815e15	0.838493	0.419247	+	0.907872i	$0.362294\pi$
$374$	8.34033e14i	0.304757i
$375$	0	0
$376$	−1.15475e15	−0.408657
$377$	4.52845e15i	1.57725i
$378$	0	0
$379$	1.80354e15	0.608542	0.304271	−	0.952586i	$-0.401587\pi$
$379$	1.80354e15	0.608542	0.304271	+	0.952586i	$0.401587\pi$
$380$	1.96645e15i	0.653101i
$381$	0	0
$382$	3.07231e15	0.988746
$383$	1.24733e15i	0.395174i	0.980285	+	0.197587i	$0.0633106\pi$
$383$	1.24733e15i	0.395174i	−0.980285	+	0.197587i	$0.936689\pi$
$384$	0	0
$385$	1.98275e14	0.0608839
$386$	1.78314e15i	0.539092i
$387$	0	0
$388$	−1.39273e15	−0.408204
$389$	3.25834e15i	0.940370i	0.882568	+	0.470185i	$0.155813\pi$
$389$	3.25834e15i	0.940370i	−0.882568	+	0.470185i	$0.844187\pi$
$390$	0	0
$391$	−2.15414e15	−0.602856
$392$	1.27320e15i	0.350896i
$393$	0	0
$394$	−9.95415e14	−0.266089
$395$	− 5.06477e15i	− 1.33345i
$396$	0	0
$397$	−4.55509e15	−1.16347	−0.581733	−	0.813379i	$-0.697625\pi$
$397$	−4.55509e15	−1.16347	−0.581733	+	0.813379i	$0.697625\pi$
$398$	3.85965e15i	0.971067i
$399$	0	0
$400$	−2.30809e14	−0.0563499
$401$	− 8.82410e14i	− 0.212229i	−0.994354	−	0.106114i	$-0.966159\pi$
$401$	− 8.82410e14i	− 0.212229i	0.994354	−	0.106114i	$-0.0338410\pi$
$402$	0	0
$403$	−8.74813e14	−0.204214
$404$	− 4.82084e14i	− 0.110875i
$405$	0	0
$406$	−3.02725e14	−0.0675916
$407$	− 4.83347e15i	− 1.06339i
$408$	0	0
$409$	2.00802e15	0.428971	0.214485	−	0.976727i	$-0.431193\pi$
$409$	2.00802e15	0.428971	0.214485	+	0.976727i	$0.431193\pi$
$410$	5.99684e15i	1.26247i
$411$	0	0
$412$	−1.43238e15	−0.292870
$413$	3.94068e14i	0.0794091i
$414$	0	0
$415$	−6.62516e15	−1.29690
$416$	1.31055e15i	0.252867i
$417$	0	0
$418$	−2.82361e15	−0.529355
$419$	− 2.91395e14i	− 0.0538515i	−0.999637	−	0.0269258i	$-0.991428\pi$
$419$	− 2.91395e14i	− 0.0538515i	0.999637	−	0.0269258i	$-0.00857177\pi$
$420$	0	0
$421$	6.13321e15	1.10153	0.550764	−	0.834661i	$-0.314337\pi$
$421$	6.13321e15	1.10153	0.550764	+	0.834661i	$0.314337\pi$
$422$	5.47836e15i	0.970008i
$423$	0	0
$424$	1.46898e15	0.252825
$425$	9.02328e14i	0.153120i
$426$	0	0
$427$	−4.63427e13	−0.00764564
$428$	− 5.79930e15i	− 0.943437i
$429$	0	0
$430$	4.92973e15	0.779852
$431$	− 6.99147e15i	− 1.09070i	−0.838209	−	0.545350i	$-0.816397\pi$
$431$	− 6.99147e15i	− 1.09070i	0.838209	−	0.545350i	$-0.183603\pi$
$432$	0	0
$433$	7.48821e15	1.13619	0.568095	−	0.822963i	$-0.307681\pi$
$433$	7.48821e15	1.13619	0.568095	+	0.822963i	$0.307681\pi$
$434$	− 5.84809e13i	− 0.00875136i
$435$	0	0
$436$	−4.39235e15	−0.639408
$437$	− 7.29282e15i	− 1.04715i
$438$	0	0
$439$	−1.03582e16	−1.44710	−0.723549	−	0.690273i	$-0.757490\pi$
$439$	−1.03582e16	−1.44710	−0.723549	+	0.690273i	$0.757490\pi$
$440$	− 1.80178e15i	− 0.248305i
$441$	0	0
$442$	5.12347e15	0.687116
$443$	− 1.03121e16i	− 1.36435i	−0.731188	−	0.682176i	$-0.761034\pi$
$443$	− 1.03121e16i	− 1.36435i	0.731188	−	0.682176i	$-0.238966\pi$
$444$	0	0
$445$	−8.71013e15	−1.12167
$446$	− 2.89971e15i	− 0.368422i
$447$	0	0
$448$	−8.76095e13	−0.0108364
$449$	− 8.52193e15i	− 1.04006i	−0.854147	−	0.520032i	$-0.825920\pi$
$449$	− 8.52193e15i	− 1.04006i	0.854147	−	0.520032i	$-0.174080\pi$
$450$	0	0
$451$	−8.61085e15	−1.02326
$452$	− 3.81001e13i	− 0.00446782i
$453$	0	0
$454$	5.28653e14	0.0603720
$455$	− 1.21800e15i	− 0.137271i
$456$	0	0
$457$	6.24229e15	0.685247	0.342623	−	0.939473i	$-0.388685\pi$
$457$	6.24229e15	0.685247	0.342623	+	0.939473i	$0.388685\pi$
$458$	6.11052e15i	0.662042i
$459$	0	0
$460$	4.65362e15	0.491184
$461$	− 6.44462e15i	− 0.671416i	−0.941966	−	0.335708i	$-0.891024\pi$
$461$	− 6.44462e15i	− 0.671416i	0.941966	−	0.335708i	$-0.108976\pi$
$462$	0	0
$463$	1.14884e16	1.16620	0.583099	−	0.812401i	$-0.301840\pi$
$463$	1.14884e16	1.16620	0.583099	+	0.812401i	$0.301840\pi$
$464$	2.75095e15i	0.275660i
$465$	0	0
$466$	3.23926e15	0.316323
$467$	− 5.88763e15i	− 0.567596i	−0.958884	−	0.283798i	$-0.908406\pi$
$467$	− 5.88763e15i	− 0.567596i	0.958884	−	0.283798i	$-0.0915944\pi$
$468$	0	0
$469$	−8.15076e14	−0.0765880
$470$	− 9.75249e15i	− 0.904749i
$471$	0	0
$472$	3.58100e15	0.323856
$473$	7.07858e15i	0.632090i
$474$	0	0
$475$	−3.05482e15	−0.265965
$476$	3.42502e14i	0.0294456i
$477$	0	0
$478$	9.62779e15	0.807159
$479$	− 7.95956e15i	− 0.658986i	−0.944158	−	0.329493i	$-0.893122\pi$
$479$	− 7.95956e15i	− 0.658986i	0.944158	−	0.329493i	$-0.106878\pi$
$480$	0	0
$481$	−2.96920e16	−2.39756
$482$	9.87445e15i	0.787464i
$483$	0	0
$484$	−3.84034e15	−0.298743
$485$	− 1.17624e16i	− 0.903746i
$486$	0	0
$487$	1.24329e16	0.931962	0.465981	−	0.884795i	$-0.345702\pi$
$487$	1.24329e16	0.931962	0.465981	+	0.884795i	$0.345702\pi$
$488$	4.21129e14i	0.0311814i
$489$	0	0
$490$	−1.07529e16	−0.776869
$491$	1.26618e16i	0.903662i	0.892103	+	0.451831i	$0.149229\pi$
$491$	1.26618e16i	0.903662i	−0.892103	+	0.451831i	$0.850771\pi$
$492$	0	0
$493$	1.07546e16	0.749053
$494$	1.73454e16i	1.19350i
$495$	0	0
$496$	−5.31432e14	−0.0356909
$497$	− 8.20432e14i	− 0.0544383i
$498$	0	0
$499$	2.62146e16	1.69801	0.849006	−	0.528384i	$-0.177202\pi$
$499$	2.62146e16	1.69801	0.849006	+	0.528384i	$0.177202\pi$
$500$	6.69895e15i	0.428733i
$501$	0	0
$502$	1.00730e16	0.629412
$503$	1.31502e16i	0.811939i	0.913887	+	0.405970i	$0.133066\pi$
$503$	1.31502e16i	0.811939i	−0.913887	+	0.405970i	$0.866934\pi$
$504$	0	0
$505$	4.07148e15	0.245473
$506$	6.68212e15i	0.398117i
$507$	0	0
$508$	8.57856e15	0.499151
$509$	− 1.89012e16i	− 1.08689i	−0.839446	−	0.543443i	$-0.817121\pi$
$509$	− 1.89012e16i	− 1.08689i	0.839446	−	0.543443i	$-0.182879\pi$
$510$	0	0
$511$	2.90843e15	0.163355
$512$	7.96131e14i	0.0441942i
$513$	0	0
$514$	−3.77730e15	−0.204834
$515$	− 1.20973e16i	− 0.648401i
$516$	0	0
$517$	1.40036e16	0.733323
$518$	− 1.98490e15i	− 0.102745i
$519$	0	0
$520$	−1.10683e16	−0.559837
$521$	− 1.63867e16i	− 0.819341i	−0.912234	−	0.409670i	$-0.865644\pi$
$521$	− 1.63867e16i	− 0.819341i	0.912234	−	0.409670i	$-0.134356\pi$
$522$	0	0
$523$	−4.73604e15	−0.231422	−0.115711	−	0.993283i	$-0.536915\pi$
$523$	−4.73604e15	−0.231422	−0.115711	+	0.993283i	$0.536915\pi$
$524$	1.03202e16i	0.498542i
$525$	0	0
$526$	−5.80911e15	−0.274281
$527$	2.07759e15i	0.0969830i
$528$	0	0
$529$	4.65604e15	0.212463
$530$	1.24063e16i	0.559743i
$531$	0	0
$532$	−1.15954e15	−0.0511463
$533$	5.28964e16i	2.30708i
$534$	0	0
$535$	4.89784e16	2.08873
$536$	7.40682e15i	0.312351i
$537$	0	0
$538$	1.36439e16	0.562660
$539$	− 1.54400e16i	− 0.629672i
$540$	0	0
$541$	−7.78326e15	−0.310440	−0.155220	−	0.987880i	$-0.549609\pi$
$541$	−7.78326e15	−0.310440	−0.155220	+	0.987880i	$0.549609\pi$
$542$	2.14069e15i	0.0844418i
$543$	0	0
$544$	3.11241e15	0.120089
$545$	− 3.70959e16i	− 1.41562i
$546$	0	0
$547$	2.37838e16	0.887887	0.443944	−	0.896055i	$-0.353579\pi$
$547$	2.37838e16	0.887887	0.443944	+	0.896055i	$0.353579\pi$
$548$	1.15978e16i	0.428245i
$549$	0	0
$550$	2.79901e15	0.101118
$551$	3.64096e16i	1.30109i
$552$	0	0
$553$	2.98650e15	0.104427
$554$	− 2.87922e16i	− 0.995903i
$555$	0	0
$556$	−5.24586e15	−0.177569
$557$	2.51025e16i	0.840594i	0.907387	+	0.420297i	$0.138074\pi$
$557$	2.51025e16i	0.840594i	−0.907387	+	0.420297i	$0.861926\pi$
$558$	0	0
$559$	4.34837e16	1.42513
$560$	− 7.39912e14i	− 0.0239912i
$561$	0	0
$562$	2.10545e16	0.668232
$563$	6.86357e15i	0.215526i	0.994177	+	0.107763i	$0.0343687\pi$
$563$	6.86357e15i	0.215526i	−0.994177	+	0.107763i	$0.965631\pi$
$564$	0	0
$565$	3.21777e14	0.00989156
$566$	− 3.27244e16i	− 0.995345i
$567$	0	0
$568$	−7.45549e15	−0.222017
$569$	− 2.50402e16i	− 0.737844i	−0.929460	−	0.368922i	$-0.879727\pi$
$569$	− 2.50402e16i	− 0.737844i	0.929460	−	0.368922i	$-0.120273\pi$
$570$	0	0
$571$	3.44263e16	0.993286	0.496643	−	0.867955i	$-0.334566\pi$
$571$	3.44263e16	0.993286	0.496643	+	0.867955i	$0.334566\pi$
$572$	− 1.58929e16i	− 0.453762i
$573$	0	0
$574$	−3.53611e15	−0.0988676
$575$	7.22929e15i	0.200027i
$576$	0	0
$577$	2.96094e15	0.0802370	0.0401185	−	0.999195i	$-0.487226\pi$
$577$	2.96094e15	0.0802370	0.0401185	+	0.999195i	$0.487226\pi$
$578$	1.41988e16i	0.380789i
$579$	0	0
$580$	−2.32333e16	−0.610300
$581$	− 3.90660e15i	− 0.101565i
$582$	0	0
$583$	−1.78142e16	−0.453686
$584$	− 2.64297e16i	− 0.666216i
$585$	0	0
$586$	1.16698e16	0.288190
$587$	− 4.45061e16i	− 1.08790i	−0.839116	−	0.543952i	$-0.816927\pi$
$587$	− 4.45061e16i	− 1.08790i	0.839116	−	0.543952i	$-0.183073\pi$
$588$	0	0
$589$	−7.03365e15	−0.168457
$590$	3.02436e16i	0.717003i
$591$	0	0
$592$	−1.80373e16	−0.419027
$593$	7.91776e16i	1.82085i	0.413673	+	0.910426i	$0.364246\pi$
$593$	7.91776e16i	1.82085i	−0.413673	+	0.910426i	$0.635754\pi$
$594$	0	0
$595$	−2.89262e15	−0.0651913
$596$	4.15845e16i	0.927799i
$597$	0	0
$598$	4.10483e16	0.897611
$599$	7.33502e16i	1.58796i	0.607941	+	0.793982i	$0.291996\pi$
$599$	7.33502e16i	1.58796i	−0.607941	+	0.793982i	$0.708004\pi$
$600$	0	0
$601$	9.33134e16	1.98015	0.990073	−	0.140553i	$-0.0448882\pi$
$601$	9.33134e16	1.98015	0.990073	+	0.140553i	$0.0448882\pi$
$602$	2.90687e15i	0.0610726i
$603$	0	0
$604$	−5.50722e15	−0.113425
$605$	− 3.24339e16i	− 0.661403i
$606$	0	0
$607$	−7.10899e15	−0.142127	−0.0710634	−	0.997472i	$-0.522639\pi$
$607$	−7.10899e15	−0.142127	−0.0710634	+	0.997472i	$0.522639\pi$
$608$	1.05370e16i	0.208591i
$609$	0	0
$610$	−3.55667e15	−0.0690343
$611$	− 8.60239e16i	− 1.65338i
$612$	0	0
$613$	2.31857e16	0.436975	0.218488	−	0.975840i	$-0.429888\pi$
$613$	2.31857e16	0.436975	0.218488	+	0.975840i	$0.429888\pi$
$614$	− 2.80471e16i	− 0.523453i
$615$	0	0
$616$	1.06244e15	0.0194455
$617$	− 6.39265e16i	− 1.15870i	−0.815080	−	0.579348i	$-0.803307\pi$
$617$	− 6.39265e16i	− 1.15870i	0.815080	−	0.579348i	$-0.196693\pi$
$618$	0	0
$619$	−1.71305e16	−0.304528	−0.152264	−	0.988340i	$-0.548656\pi$
$619$	−1.71305e16	−0.304528	−0.152264	+	0.988340i	$0.548656\pi$
$620$	− 4.48825e15i	− 0.0790181i
$621$	0	0
$622$	−3.98952e16	−0.688936
$623$	− 5.13602e15i	− 0.0878413i
$624$	0	0
$625$	−7.00113e16	−1.17459
$626$	6.39232e16i	1.06222i
$627$	0	0
$628$	4.28552e16	0.698628
$629$	7.05153e16i	1.13862i
$630$	0	0
$631$	6.80683e16	1.07837	0.539186	−	0.842187i	$-0.318732\pi$
$631$	6.80683e16	1.07837	0.539186	+	0.842187i	$0.318732\pi$
$632$	− 2.71391e16i	− 0.425886i
$633$	0	0
$634$	6.16228e16	0.948868
$635$	7.24508e16i	1.10510i
$636$	0	0
$637$	−9.48479e16	−1.41968
$638$	− 3.33606e16i	− 0.494664i
$639$	0	0
$640$	−6.72378e15	−0.0978439
$641$	4.56923e15i	0.0658711i	0.999457	+	0.0329355i	$0.0104856\pi$
$641$	4.56923e15i	0.0658711i	−0.999457	+	0.0329355i	$0.989514\pi$
$642$	0	0
$643$	−1.28989e17	−1.82510	−0.912549	−	0.408966i	$-0.865889\pi$
$643$	−1.28989e17	−1.82510	−0.912549	+	0.408966i	$0.865889\pi$
$644$	2.74406e15i	0.0384661i
$645$	0	0
$646$	4.11936e16	0.566806
$647$	8.35683e16i	1.13924i	0.821907	+	0.569621i	$0.192910\pi$
$647$	8.35683e16i	1.13924i	−0.821907	+	0.569621i	$0.807090\pi$
$648$	0	0
$649$	−4.34266e16	−0.581150
$650$	− 1.71943e16i	− 0.227985i
$651$	0	0
$652$	−3.87326e16	−0.504186
$653$	− 2.95494e16i	− 0.381127i	−0.981675	−	0.190564i	$-0.938968\pi$
$653$	− 2.95494e16i	− 0.381127i	0.981675	−	0.190564i	$-0.0610316\pi$
$654$	0	0
$655$	−8.71603e16	−1.10375
$656$	3.21336e16i	0.403214i
$657$	0	0
$658$	5.75066e15	0.0708537
$659$	5.33291e16i	0.651106i	0.945524	+	0.325553i	$0.105550\pi$
$659$	5.33291e16i	0.651106i	−0.945524	+	0.325553i	$0.894450\pi$
$660$	0	0
$661$	−1.25781e17	−1.50802	−0.754009	−	0.656865i	$-0.771882\pi$
$661$	−1.25781e17	−1.50802	−0.754009	+	0.656865i	$0.771882\pi$
$662$	− 8.61077e16i	− 1.02304i
$663$	0	0
$664$	−3.55003e16	−0.414213
$665$	− 9.79295e15i	− 0.113236i
$666$	0	0
$667$	8.61638e16	0.978521
$668$	− 5.79382e16i	− 0.652088i
$669$	0	0
$670$	−6.25548e16	−0.691531
$671$	− 5.10701e15i	− 0.0559541i
$672$	0	0
$673$	1.91737e16	0.206355	0.103178	−	0.994663i	$-0.467099\pi$
$673$	1.91737e16	0.206355	0.103178	+	0.994663i	$0.467099\pi$
$674$	− 9.80344e16i	− 1.04573i
$675$	0	0
$676$	−4.99159e16	−0.523069
$677$	4.64044e16i	0.481978i	0.970528	+	0.240989i	$0.0774717\pi$
$677$	4.64044e16i	0.481978i	−0.970528	+	0.240989i	$0.922528\pi$
$678$	0	0
$679$	6.93584e15	0.0707751
$680$	2.62860e16i	0.265871i
$681$	0	0
$682$	6.44465e15	0.0640462
$683$	− 8.17955e16i	− 0.805759i	−0.915253	−	0.402879i	$-0.868009\pi$
$683$	− 8.17955e16i	− 0.805759i	0.915253	−	0.402879i	$-0.131991\pi$
$684$	0	0
$685$	−9.79501e16	−0.948116
$686$	− 1.27291e16i	− 0.122139i
$687$	0	0
$688$	2.64155e16	0.249074
$689$	1.09433e17i	1.02290i
$690$	0	0
$691$	1.03674e17	0.952358	0.476179	−	0.879348i	$-0.342021\pi$
$691$	1.03674e17	0.952358	0.476179	+	0.879348i	$0.342021\pi$
$692$	− 2.52377e16i	− 0.229833i
$693$	0	0
$694$	2.90264e16	0.259799
$695$	− 4.43043e16i	− 0.393131i
$696$	0	0
$697$	1.25623e17	1.09565
$698$	− 5.47788e16i	− 0.473675i
$699$	0	0
$700$	1.14944e15	0.00977004
$701$	1.01644e16i	0.0856590i	0.999082	+	0.0428295i	$0.0136372\pi$
$701$	1.01644e16i	0.0856590i	−0.999082	+	0.0428295i	$0.986363\pi$
$702$	0	0
$703$	−2.38729e17	−1.97776
$704$	− 9.65465e15i	− 0.0793050i
$705$	0	0
$706$	3.93117e16	0.317463
$707$	2.40079e15i	0.0192237i
$708$	0	0
$709$	1.00095e17	0.788016	0.394008	−	0.919107i	$-0.371088\pi$
$709$	1.00095e17	0.788016	0.394008	+	0.919107i	$0.371088\pi$
$710$	− 6.29658e16i	− 0.491536i
$711$	0	0
$712$	−4.66724e16	−0.358245
$713$	1.66452e16i	0.126693i
$714$	0	0
$715$	1.34225e17	1.00461
$716$	− 8.71312e16i	− 0.646689i
$717$	0	0
$718$	−7.38825e16	−0.539256
$719$	4.11390e16i	0.297769i	0.988855	+	0.148885i	$0.0475683\pi$
$719$	4.11390e16i	0.297769i	−0.988855	+	0.148885i	$0.952432\pi$
$720$	0	0
$721$	7.13328e15	0.0507782
$722$	3.92972e16i	0.277421i
$723$	0	0
$724$	−7.99149e16	−0.554877
$725$	− 3.60924e16i	− 0.248535i
$726$	0	0
$727$	−2.10764e17	−1.42755	−0.713774	−	0.700376i	$-0.753015\pi$
$727$	−2.10764e17	−1.42755	−0.713774	+	0.700376i	$0.753015\pi$
$728$	− 6.52655e15i	− 0.0438425i
$729$	0	0
$730$	2.23214e17	1.47497
$731$	− 1.03269e17i	− 0.676809i
$732$	0	0
$733$	−1.81932e17	−1.17297	−0.586483	−	0.809962i	$-0.699488\pi$
$733$	−1.81932e17	−1.17297	−0.586483	+	0.809962i	$0.699488\pi$
$734$	− 1.55537e17i	− 0.994622i
$735$	0	0
$736$	2.49360e16	0.156877
$737$	− 8.98222e16i	− 0.560504i
$738$	0	0
$739$	−2.23483e17	−1.37207	−0.686037	−	0.727567i	$-0.740651\pi$
$739$	−2.23483e17	−1.37207	−0.686037	+	0.727567i	$0.740651\pi$
$740$	− 1.52335e17i	− 0.927706i
$741$	0	0
$742$	−7.31554e15	−0.0438352
$743$	− 1.47865e17i	− 0.878884i	−0.898271	−	0.439442i	$-0.855176\pi$
$743$	− 1.47865e17i	− 0.878884i	0.898271	−	0.439442i	$-0.144824\pi$
$744$	0	0
$745$	−3.51205e17	−2.05411
$746$	1.02192e17i	0.592904i
$747$	0	0
$748$	−3.77440e16	−0.215496
$749$	2.88807e16i	0.163575i
$750$	0	0
$751$	4.85677e16	0.270712	0.135356	−	0.990797i	$-0.456782\pi$
$751$	4.85677e16	0.270712	0.135356	+	0.990797i	$0.456782\pi$
$752$	− 5.22578e16i	− 0.288964i
$753$	0	0
$754$	−2.04934e17	−1.11529
$755$	− 4.65116e16i	− 0.251119i
$756$	0	0
$757$	4.14209e16	0.220112	0.110056	−	0.993925i	$-0.464897\pi$
$757$	4.14209e16	0.220112	0.110056	+	0.993925i	$0.464897\pi$
$758$	8.16189e16i	0.430304i
$759$	0	0
$760$	−8.89911e16	−0.461812
$761$	− 1.84226e17i	− 0.948514i	−0.880386	−	0.474257i	$-0.842717\pi$
$761$	− 1.84226e17i	− 0.948514i	0.880386	−	0.474257i	$-0.157283\pi$
$762$	0	0
$763$	2.18740e16	0.110862
$764$	1.39037e17i	0.699149i
$765$	0	0
$766$	−5.64477e16	−0.279431
$767$	2.66770e17i	1.31028i
$768$	0	0
$769$	2.04287e17	0.987832	0.493916	−	0.869510i	$-0.335565\pi$
$769$	2.04287e17	0.987832	0.493916	+	0.869510i	$0.335565\pi$
$770$	8.97289e15i	0.0430515i
$771$	0	0
$772$	−8.06959e16	−0.381196
$773$	8.18279e16i	0.383552i	0.981439	+	0.191776i	$0.0614248\pi$
$773$	8.18279e16i	0.383552i	−0.981439	+	0.191776i	$0.938575\pi$
$774$	0	0
$775$	6.97238e15	0.0321789
$776$	− 6.30279e16i	− 0.288644i
$777$	0	0
$778$	−1.47456e17	−0.664942
$779$	4.25297e17i	1.90312i
$780$	0	0
$781$	9.04124e16	0.398402
$782$	− 9.74852e16i	− 0.426283i
$783$	0	0
$784$	−5.76183e16	−0.248121
$785$	3.61937e17i	1.54673i
$786$	0	0
$787$	2.40385e17	1.01172	0.505858	−	0.862617i	$-0.331176\pi$
$787$	2.40385e17	1.01172	0.505858	+	0.862617i	$0.331176\pi$
$788$	− 4.50473e16i	− 0.188153i
$789$	0	0
$790$	2.29206e17	0.942893
$791$	1.89740e14i	0 0.000774639i
$792$	0	0
$793$	−3.13724e16	−0.126156
$794$	− 2.06140e17i	− 0.822695i
$795$	0	0
$796$	−1.74668e17	−0.686648
$797$	3.75671e17i	1.46574i	0.680367	+	0.732872i	$0.261821\pi$
$797$	3.75671e17i	1.46574i	−0.680367	+	0.732872i	$0.738179\pi$
$798$	0	0
$799$	−2.04297e17	−0.785204
$800$	− 1.04452e16i	− 0.0398454i
$801$	0	0
$802$	3.99333e16	0.150068
$803$	3.20512e17i	1.19550i
$804$	0	0
$805$	−2.31752e16	−0.0851623
$806$	− 3.95895e16i	− 0.144401i
$807$	0	0
$808$	2.18166e16	0.0784007
$809$	− 1.31094e16i	− 0.0467618i	−0.999727	−	0.0233809i	$-0.992557\pi$
$809$	− 1.31094e16i	− 0.0467618i	0.999727	−	0.0233809i	$-0.00744304\pi$
$810$	0	0
$811$	−2.23462e16	−0.0785379	−0.0392690	−	0.999229i	$-0.512503\pi$
$811$	−2.23462e16	−0.0785379	−0.0392690	+	0.999229i	$0.512503\pi$
$812$	− 1.36998e16i	− 0.0477945i
$813$	0	0
$814$	2.18738e17	0.751930
$815$	− 3.27119e17i	− 1.11625i
$816$	0	0
$817$	3.49617e17	1.17560
$818$	9.08725e16i	0.303328i
$819$	0	0
$820$	−2.71386e17	−0.892698
$821$	1.00359e15i	0.00327715i	0.999999	+	0.00163857i	$0.000521575\pi$
$821$	1.00359e15i	0.00327715i	−0.999999	+	0.00163857i	$0.999478\pi$
$822$	0	0
$823$	−4.29591e17	−1.38247	−0.691235	−	0.722630i	$-0.742933\pi$
$823$	−4.29591e17	−1.38247	−0.691235	+	0.722630i	$0.742933\pi$
$824$	− 6.48221e16i	− 0.207090i
$825$	0	0
$826$	−1.78335e16	−0.0561507
$827$	5.28099e17i	1.65075i	0.564583	+	0.825376i	$0.309037\pi$
$827$	5.28099e17i	1.65075i	−0.564583	+	0.825376i	$0.690963\pi$
$828$	0	0
$829$	1.29706e16	0.0399605	0.0199803	−	0.999800i	$-0.493640\pi$
$829$	1.29706e16	0.0399605	0.0199803	+	0.999800i	$0.493640\pi$
$830$	− 2.99821e17i	− 0.917050i
$831$	0	0
$832$	−5.93086e16	−0.178804
$833$	2.25254e17i	0.674220i
$834$	0	0
$835$	4.89321e17	1.44369
$836$	− 1.27782e17i	− 0.374311i
$837$	0	0
$838$	1.31870e16	0.0380788
$839$	− 3.22584e17i	− 0.924849i	−0.886659	−	0.462425i	$-0.846980\pi$
$839$	− 3.22584e17i	− 0.924849i	0.886659	−	0.462425i	$-0.153020\pi$
$840$	0	0
$841$	−7.63599e16	−0.215819
$842$	2.77557e17i	0.778898i
$843$	0	0
$844$	−2.47922e17	−0.685900
$845$	− 4.21568e17i	− 1.15805i
$846$	0	0
$847$	1.91250e16	0.0517965
$848$	6.64783e16i	0.178774i
$849$	0	0
$850$	−4.08347e16	−0.108272
$851$	5.64956e17i	1.48743i
$852$	0	0
$853$	1.22272e17	0.317419	0.158709	−	0.987325i	$-0.449267\pi$
$853$	1.22272e17	0.317419	0.158709	+	0.987325i	$0.449267\pi$
$854$	− 2.09723e15i	− 0.00540629i
$855$	0	0
$856$	2.62446e17	0.667111
$857$	− 1.10840e17i	− 0.279776i	−0.990167	−	0.139888i	$-0.955326\pi$
$857$	− 1.10840e17i	− 0.279776i	0.990167	−	0.139888i	$-0.0446742\pi$
$858$	0	0
$859$	−4.31456e17	−1.07393	−0.536967	−	0.843603i	$-0.680430\pi$
$859$	−4.31456e17	−1.07393	−0.536967	+	0.843603i	$0.680430\pi$
$860$	2.23094e17i	0.551439i
$861$	0	0
$862$	3.16398e17	0.771241
$863$	4.77620e17i	1.15616i	0.815980	+	0.578079i	$0.196198\pi$
$863$	4.77620e17i	1.15616i	−0.815980	+	0.578079i	$0.803802\pi$
$864$	0	0
$865$	2.13147e17	0.508841
$866$	3.38878e17i	0.803407i
$867$	0	0
$868$	2.64654e15	0.00618815
$869$	3.29115e17i	0.764239i
$870$	0	0
$871$	−5.51778e17	−1.26373
$872$	− 1.98775e17i	− 0.452130i
$873$	0	0
$874$	3.30035e17	0.740443
$875$	− 3.33609e16i	− 0.0743344i
$876$	0	0
$877$	−2.20897e17	−0.485502	−0.242751	−	0.970089i	$-0.578050\pi$
$877$	−2.20897e17	−0.485502	−0.242751	+	0.970089i	$0.578050\pi$
$878$	− 4.68759e17i	− 1.02325i
$879$	0	0
$880$	8.15390e16	0.175578
$881$	9.40022e16i	0.201040i	0.994935	+	0.100520i	$0.0320506\pi$
$881$	9.40022e16i	0.201040i	−0.994935	+	0.100520i	$0.967949\pi$
$882$	0	0
$883$	3.12675e17	0.659673	0.329836	−	0.944038i	$-0.393006\pi$
$883$	3.12675e17	0.659673	0.329836	+	0.944038i	$0.393006\pi$
$884$	2.31862e17i	0.485865i
$885$	0	0
$886$	4.66674e17	0.964742
$887$	− 1.61036e17i	− 0.330659i	−0.986238	−	0.165330i	$-0.947131\pi$
$887$	− 1.61036e17i	− 0.330659i	0.986238	−	0.165330i	$-0.0528688\pi$
$888$	0	0
$889$	−4.27214e16	−0.0865436
$890$	− 3.94175e17i	− 0.793139i
$891$	0	0
$892$	1.31226e17	0.260513
$893$	− 6.91647e17i	− 1.36388i
$894$	0	0
$895$	7.35873e17	1.43174
$896$	− 3.96475e15i	− 0.00766246i
$897$	0	0
$898$	3.85659e17	0.735436
$899$	− 8.31018e16i	− 0.157417i
$900$	0	0
$901$	2.59891e17	0.485783
$902$	− 3.89682e17i	− 0.723555i
$903$	0	0
$904$	1.72422e15	0.00315923
$905$	− 6.74927e17i	− 1.22847i
$906$	0	0
$907$	2.25307e17	0.404698	0.202349	−	0.979314i	$-0.435143\pi$
$907$	2.25307e17	0.404698	0.202349	+	0.979314i	$0.435143\pi$
$908$	2.39241e16i	0.0426895i
$909$	0	0
$910$	5.51205e16	0.0970654
$911$	− 2.32226e17i	− 0.406257i	−0.979152	−	0.203128i	$-0.934889\pi$
$911$	− 2.32226e17i	− 0.406257i	0.979152	−	0.203128i	$-0.0651109\pi$
$912$	0	0
$913$	4.30511e17	0.743292
$914$	2.82494e17i	0.484543i
$915$	0	0
$916$	−2.76531e17	−0.468134
$917$	− 5.13950e16i	− 0.0864381i
$918$	0	0
$919$	8.95398e17	1.48636	0.743179	−	0.669093i	$-0.233317\pi$
$919$	8.95398e17	1.48636	0.743179	+	0.669093i	$0.233317\pi$
$920$	2.10599e17i	0.347320i
$921$	0	0
$922$	2.91650e17	0.474763
$923$	− 5.55404e17i	− 0.898253i
$924$	0	0
$925$	2.36649e17	0.377794
$926$	5.19904e17i	0.824626i
$927$	0	0
$928$	−1.24494e17	−0.194921
$929$	− 1.99521e17i	− 0.310381i	−0.987885	−	0.155190i	$-0.950401\pi$
$929$	− 1.99521e17i	− 0.310381i	0.987885	−	0.155190i	$-0.0495991\pi$
$930$	0	0
$931$	−7.62594e17	−1.17110
$932$	1.46592e17i	0.223674i
$933$	0	0
$934$	2.66444e17	0.401351
$935$	− 3.18770e17i	− 0.477098i
$936$	0	0
$937$	4.44483e17	0.656777	0.328389	−	0.944543i	$-0.393494\pi$
$937$	4.44483e17	0.656777	0.328389	+	0.944543i	$0.393494\pi$
$938$	− 3.68861e16i	− 0.0541559i
$939$	0	0
$940$	4.41347e17	0.639754
$941$	− 3.98567e17i	− 0.574068i	−0.957920	−	0.287034i	$-0.907331\pi$
$941$	− 3.98567e17i	− 0.574068i	0.957920	−	0.287034i	$-0.0926693\pi$
$942$	0	0
$943$	1.00647e18	1.43130
$944$	1.62058e17i	0.229001i
$945$	0	0
$946$	−3.20340e17	−0.446955
$947$	6.55940e17i	0.909420i	0.890640	+	0.454710i	$0.150257\pi$
$947$	6.55940e17i	0.909420i	−0.890640	+	0.454710i	$0.849743\pi$
$948$	0	0
$949$	1.96891e18	2.69543
$950$	− 1.38246e17i	− 0.188066i
$951$	0	0
$952$	−1.54999e16	−0.0208212
$953$	2.85817e17i	0.381532i	0.981636	+	0.190766i	$0.0610972\pi$
$953$	2.85817e17i	0.381532i	−0.981636	+	0.190766i	$0.938903\pi$
$954$	0	0
$955$	−1.17425e18	−1.54788
$956$	4.35704e17i	0.570748i
$957$	0	0
$958$	3.60209e17	0.465973
$959$	− 5.77574e16i	− 0.0742499i
$960$	0	0
$961$	−7.71609e17	−0.979619
$962$	− 1.34371e18i	− 1.69533i
$963$	0	0
$964$	−4.46867e17	−0.556821
$965$	− 6.81523e17i	− 0.843950i
$966$	0	0
$967$	−9.51678e17	−1.16394	−0.581971	−	0.813210i	$-0.697718\pi$
$967$	−9.51678e17	−1.16394	−0.581971	+	0.813210i	$0.697718\pi$
$968$	− 1.73794e17i	− 0.211243i
$969$	0	0
$970$	5.32306e17	0.639045
$971$	6.07274e16i	0.0724552i	0.999344	+	0.0362276i	$0.0115341\pi$
$971$	6.07274e16i	0.0724552i	−0.999344	+	0.0362276i	$0.988466\pi$
$972$	0	0
$973$	2.61245e16	0.0307873
$974$	5.62648e17i	0.658996i
$975$	0	0
$976$	−1.90581e16	−0.0220486
$977$	1.17338e17i	0.134919i	0.997722	+	0.0674594i	$0.0214893\pi$
$977$	1.17338e17i	0.134919i	−0.997722	+	0.0674594i	$0.978511\pi$
$978$	0	0
$979$	5.65995e17	0.642860
$980$	− 4.86619e17i	− 0.549329i
$981$	0	0
$982$	−5.73007e17	−0.638986
$983$	1.41778e18i	1.57141i	0.618602	+	0.785704i	$0.287699\pi$
$983$	1.41778e18i	1.57141i	−0.618602	+	0.785704i	$0.712301\pi$
$984$	0	0
$985$	3.80450e17	0.416563
$986$	4.86697e17i	0.529660i
$987$	0	0
$988$	−7.84965e17	−0.843935
$989$	− 8.27373e17i	− 0.884146i
$990$	0	0
$991$	−3.75903e16	−0.0396857	−0.0198428	−	0.999803i	$-0.506317\pi$
$991$	−3.75903e16	−0.0396857	−0.0198428	+	0.999803i	$0.506317\pi$
$992$	− 2.40499e16i	− 0.0252373i
$993$	0	0
$994$	3.71285e16	0.0384937
$995$	− 1.47517e18i	− 1.52021i
$996$	0	0
$997$	−4.16797e17	−0.424378	−0.212189	−	0.977229i	$-0.568059\pi$
$997$	−4.16797e17	−0.424378	−0.212189	+	0.977229i	$0.568059\pi$
$998$	1.18634e18i	1.20068i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.13.b.c.161.15		24
3.2	odd	2	inner	162.13.b.c.161.10		24
9.2	odd	6		54.13.d.a.17.3		24
9.4	even	3		54.13.d.a.35.3		24
9.5	odd	6		18.13.d.a.11.12	yes	24
9.7	even	3		18.13.d.a.5.12	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.13.d.a.5.12	✓	24	9.7	even	3
18.13.d.a.11.12	yes	24	9.5	odd	6
54.13.d.a.17.3		24	9.2	odd	6
54.13.d.a.35.3		24	9.4	even	3
162.13.b.c.161.10		24	3.2	odd	2	inner
162.13.b.c.161.15		24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	162.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+45.2548i q^{2} -2048.00 q^{4} -17296.5i q^{5} +10199.1 q^{7} -92681.9i q^{8} +782751. q^{10} +1.12395e6i q^{11} +6.90443e6 q^{13} +461558. i q^{14} +4.19430e6 q^{16} -1.63973e7i q^{17} +5.55128e7 q^{19} +3.54233e7i q^{20} -5.08642e7 q^{22} -1.31372e8i q^{23} -5.50292e7 q^{25} +3.12459e8i q^{26} -2.08877e7 q^{28} +6.55877e8i q^{29} -1.26703e8 q^{31} +1.89813e8i q^{32} +7.42055e8 q^{34} -1.76409e8i q^{35} -4.30043e9 q^{37} +2.51222e9i q^{38} -1.60307e9 q^{40} +7.66124e9i q^{41} +6.29795e9 q^{43} -2.30185e9i q^{44} +5.94521e9 q^{46} -1.24592e10i q^{47} -1.37373e10 q^{49} -2.49034e9i q^{50} -1.41403e10 q^{52} +1.58497e10i q^{53} +1.94404e10 q^{55} -9.45271e8i q^{56} -2.96816e10 q^{58} +3.86375e10i q^{59} -4.54381e9 q^{61} -5.73393e9i q^{62} -8.58993e9 q^{64} -1.19423e11i q^{65} -7.99165e10 q^{67} +3.35816e10i q^{68} +7.98335e9 q^{70} -8.04417e10i q^{71} +2.85166e11 q^{73} -1.94615e11i q^{74} -1.13690e11 q^{76} +1.14633e10i q^{77} +2.92820e11 q^{79} -7.25469e10i q^{80} -3.46708e11 q^{82} -3.83034e11i q^{83} -2.83616e11 q^{85} +2.85013e11i q^{86} +1.04170e11 q^{88} -5.03577e11i q^{89} +7.04189e10 q^{91} +2.69050e11i q^{92} +5.63841e11 q^{94} -9.60178e11i q^{95} +6.80045e11 q^{97} -6.21678e11i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 49152 q^{4} + 137280 q^{7} - 1700160 q^{13} + 100663296 q^{16} - 47432760 q^{19} + 134668800 q^{22} - 1721170488 q^{25} - 281149440 q^{28} - 2023332000 q^{31} + 755573760 q^{34} - 8737330560 q^{37}+ \cdots + 1991581150440 q^{97}+O(q^{100}) \) 24 * q - 49152 * q^4 + 137280 * q^7 - 1700160 * q^13 + 100663296 * q^16 - 47432760 * q^19 + 134668800 * q^22 - 1721170488 * q^25 - 281149440 * q^28 - 2023332000 * q^31 + 755573760 * q^34 - 8737330560 * q^37 + 2564630040 * q^43 + 16593325056 * q^46 + 76678938504 * q^49 + 3481927680 * q^52 - 82896170256 * q^55 - 2930457600 * q^58 + 144178091904 * q^61 - 206158430208 * q^64 + 364748716920 * q^67 - 150468074496 * q^70 - 404658904200 * q^73 + 97142292480 * q^76 - 255432049584 * q^79 - 966809364480 * q^82 + 341523808704 * q^85 - 275801702400 * q^88 - 563012378112 * q^91 - 311763336192 * q^94 + 1991581150440 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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