LMFDB - Embedded newform 162.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,4,Mod(1,162)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("162.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	162.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$9.55830942093$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	162.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +0.803848 q^{5} -2.39230 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} +0.803848 q^{5} -2.39230 q^{7} +8.00000 q^{8} +1.60770 q^{10} +59.5692 q^{11} +77.7461 q^{13} -4.78461 q^{14} +16.0000 q^{16} +2.84232 q^{17} -118.315 q^{19} +3.21539 q^{20} +119.138 q^{22} +110.785 q^{23} -124.354 q^{25} +155.492 q^{26} -9.56922 q^{28} +125.627 q^{29} +63.0615 q^{31} +32.0000 q^{32} +5.68465 q^{34} -1.92305 q^{35} -227.392 q^{37} -236.631 q^{38} +6.43078 q^{40} +324.631 q^{41} -272.823 q^{43} +238.277 q^{44} +221.569 q^{46} -4.93851 q^{47} -337.277 q^{49} -248.708 q^{50} +310.985 q^{52} -598.908 q^{53} +47.8846 q^{55} -19.1384 q^{56} +251.254 q^{58} -670.277 q^{59} +464.531 q^{61} +126.123 q^{62} +64.0000 q^{64} +62.4960 q^{65} -769.069 q^{67} +11.3693 q^{68} -3.84610 q^{70} -611.569 q^{71} +923.831 q^{73} -454.785 q^{74} -473.261 q^{76} -142.508 q^{77} +39.8076 q^{79} +12.8616 q^{80} +649.261 q^{82} +443.769 q^{83} +2.28479 q^{85} -545.646 q^{86} +476.554 q^{88} -78.4576 q^{89} -185.992 q^{91} +443.138 q^{92} -9.87703 q^{94} -95.1075 q^{95} -91.0155 q^{97} -674.554 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{7} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^7 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{7} + 16 q^{8} + 24 q^{10} + 36 q^{11} + 10 q^{13} + 32 q^{14} + 32 q^{16} + 120 q^{17} - 8 q^{19} + 48 q^{20} + 72 q^{22} + 180 q^{23} - 124 q^{25} + 20 q^{26} + 64 q^{28} + 324 q^{29} - 248 q^{31} + 64 q^{32} + 240 q^{34} + 204 q^{35} - 434 q^{37} - 16 q^{38} + 96 q^{40} + 192 q^{41} - 608 q^{43} + 144 q^{44} + 360 q^{46} - 384 q^{47} - 342 q^{49} - 248 q^{50} + 40 q^{52} - 408 q^{53} - 216 q^{55} + 128 q^{56} + 648 q^{58} - 1008 q^{59} + 742 q^{61} - 496 q^{62} + 128 q^{64} - 696 q^{65} - 104 q^{67} + 480 q^{68} + 408 q^{70} - 1140 q^{71} + 850 q^{73} - 868 q^{74} - 32 q^{76} - 576 q^{77} - 440 q^{79} + 192 q^{80} + 384 q^{82} + 264 q^{83} + 1314 q^{85} - 1216 q^{86} + 288 q^{88} + 768 q^{89} - 1432 q^{91} + 720 q^{92} - 768 q^{94} + 1140 q^{95} - 764 q^{97} - 684 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^7 + 16 * q^8 + 24 * q^10 + 36 * q^11 + 10 * q^13 + 32 * q^14 + 32 * q^16 + 120 * q^17 - 8 * q^19 + 48 * q^20 + 72 * q^22 + 180 * q^23 - 124 * q^25 + 20 * q^26 + 64 * q^28 + 324 * q^29 - 248 * q^31 + 64 * q^32 + 240 * q^34 + 204 * q^35 - 434 * q^37 - 16 * q^38 + 96 * q^40 + 192 * q^41 - 608 * q^43 + 144 * q^44 + 360 * q^46 - 384 * q^47 - 342 * q^49 - 248 * q^50 + 40 * q^52 - 408 * q^53 - 216 * q^55 + 128 * q^56 + 648 * q^58 - 1008 * q^59 + 742 * q^61 - 496 * q^62 + 128 * q^64 - 696 * q^65 - 104 * q^67 + 480 * q^68 + 408 * q^70 - 1140 * q^71 + 850 * q^73 - 868 * q^74 - 32 * q^76 - 576 * q^77 - 440 * q^79 + 192 * q^80 + 384 * q^82 + 264 * q^83 + 1314 * q^85 - 1216 * q^86 + 288 * q^88 + 768 * q^89 - 1432 * q^91 + 720 * q^92 - 768 * q^94 + 1140 * q^95 - 764 * q^97 - 684 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	0.803848	0.0718983	0.0359492	−	0.999354i	$-0.488555\pi$
$5$	0.803848	0.0718983	0.0359492	+	0.999354i	$0.488555\pi$
$6$	0	0
$7$	−2.39230	−0.129172	−0.0645862	−	0.997912i	$-0.520573\pi$
$7$	−2.39230	−0.129172	−0.0645862	+	0.997912i	$0.520573\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	1.60770	0.0508398
$11$	59.5692	1.63280	0.816400	−	0.577487i	$-0.195967\pi$
$11$	59.5692	1.63280	0.816400	+	0.577487i	$0.195967\pi$
$12$	0	0
$13$	77.7461	1.65868	0.829342	−	0.558741i	$-0.188715\pi$
$13$	77.7461	1.65868	0.829342	+	0.558741i	$0.188715\pi$
$14$	−4.78461	−0.0913386
$15$	0	0
$16$	16.0000	0.250000
$17$	2.84232	0.0405509	0.0202754	−	0.999794i	$-0.493546\pi$
$17$	2.84232	0.0405509	0.0202754	+	0.999794i	$0.493546\pi$
$18$	0	0
$19$	−118.315	−1.42860	−0.714300	−	0.699840i	$-0.753255\pi$
$19$	−118.315	−1.42860	−0.714300	+	0.699840i	$0.753255\pi$
$20$	3.21539	0.0359492
$21$	0	0
$22$	119.138	1.15456
$23$	110.785	1.00436	0.502178	−	0.864764i	$-0.332532\pi$
$23$	110.785	1.00436	0.502178	+	0.864764i	$0.332532\pi$
$24$	0	0
$25$	−124.354	−0.994831
$26$	155.492	1.17287
$27$	0	0
$28$	−9.56922	−0.0645862
$29$	125.627	0.804425	0.402213	−	0.915546i	$-0.368241\pi$
$29$	125.627	0.804425	0.402213	+	0.915546i	$0.368241\pi$
$30$	0	0
$31$	63.0615	0.365361	0.182680	−	0.983172i	$-0.441523\pi$
$31$	63.0615	0.365361	0.182680	+	0.983172i	$0.441523\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	5.68465	0.0286738
$35$	−1.92305	−0.00928727
$36$	0	0
$37$	−227.392	−1.01035	−0.505177	−	0.863016i	$-0.668573\pi$
$37$	−227.392	−1.01035	−0.505177	+	0.863016i	$0.668573\pi$
$38$	−236.631	−1.01017
$39$	0	0
$40$	6.43078	0.0254199
$41$	324.631	1.23656	0.618278	−	0.785959i	$-0.287831\pi$
$41$	324.631	1.23656	0.618278	+	0.785959i	$0.287831\pi$
$42$	0	0
$43$	−272.823	−0.967561	−0.483781	−	0.875189i	$-0.660737\pi$
$43$	−272.823	−0.967561	−0.483781	+	0.875189i	$0.660737\pi$
$44$	238.277	0.816400
$45$	0	0
$46$	221.569	0.710187
$47$	−4.93851	−0.0153267	−0.00766336	−	0.999971i	$-0.502439\pi$
$47$	−4.93851	−0.0153267	−0.00766336	+	0.999971i	$0.502439\pi$
$48$	0	0
$49$	−337.277	−0.983315
$50$	−248.708	−0.703451
$51$	0	0
$52$	310.985	0.829342
$53$	−598.908	−1.55219	−0.776097	−	0.630614i	$-0.782803\pi$
$53$	−598.908	−1.55219	−0.776097	+	0.630614i	$0.782803\pi$
$54$	0	0
$55$	47.8846	0.117396
$56$	−19.1384	−0.0456693
$57$	0	0
$58$	251.254	0.568815
$59$	−670.277	−1.47903	−0.739514	−	0.673142i	$-0.764944\pi$
$59$	−670.277	−1.47903	−0.739514	+	0.673142i	$0.764944\pi$
$60$	0	0
$61$	464.531	0.975034	0.487517	−	0.873114i	$-0.337903\pi$
$61$	464.531	0.975034	0.487517	+	0.873114i	$0.337903\pi$
$62$	126.123	0.258349
$63$	0	0
$64$	64.0000	0.125000
$65$	62.4960	0.119257
$66$	0	0
$67$	−769.069	−1.40234	−0.701170	−	0.712994i	$-0.747338\pi$
$67$	−769.069	−1.40234	−0.701170	+	0.712994i	$0.747338\pi$
$68$	11.3693	0.0202754
$69$	0	0
$70$	−3.84610	−0.00656709
$71$	−611.569	−1.02225	−0.511126	−	0.859506i	$-0.670772\pi$
$71$	−611.569	−1.02225	−0.511126	+	0.859506i	$0.670772\pi$
$72$	0	0
$73$	923.831	1.48118	0.740590	−	0.671957i	$-0.234546\pi$
$73$	923.831	1.48118	0.740590	+	0.671957i	$0.234546\pi$
$74$	−454.785	−0.714428
$75$	0	0
$76$	−473.261	−0.714300
$77$	−142.508	−0.210913
$78$	0	0
$79$	39.8076	0.0566925	0.0283462	−	0.999598i	$-0.490976\pi$
$79$	39.8076	0.0566925	0.0283462	+	0.999598i	$0.490976\pi$
$80$	12.8616	0.0179746
$81$	0	0
$82$	649.261	0.874377
$83$	443.769	0.586867	0.293434	−	0.955979i	$-0.405202\pi$
$83$	443.769	0.586867	0.293434	+	0.955979i	$0.405202\pi$
$84$	0	0
$85$	2.28479	0.00291554
$86$	−545.646	−0.684169
$87$	0	0
$88$	476.554	0.577282
$89$	−78.4576	−0.0934437	−0.0467218	−	0.998908i	$-0.514877\pi$
$89$	−78.4576	−0.0934437	−0.0467218	+	0.998908i	$0.514877\pi$
$90$	0	0
$91$	−185.992	−0.214256
$92$	443.138	0.502178
$93$	0	0
$94$	−9.87703	−0.0108376
$95$	−95.1075	−0.102714
$96$	0	0
$97$	−91.0155	−0.0952703	−0.0476352	−	0.998865i	$-0.515168\pi$
$97$	−91.0155	−0.0952703	−0.0476352	+	0.998865i	$0.515168\pi$
$98$	−674.554	−0.695308
$99$	0	0
$100$	−497.415	−0.497415
$101$	−1622.52	−1.59849	−0.799243	−	0.601008i	$-0.794766\pi$
$101$	−1622.52	−1.59849	−0.799243	+	0.601008i	$0.794766\pi$
$102$	0	0
$103$	−748.231	−0.715780	−0.357890	−	0.933764i	$-0.616504\pi$
$103$	−748.231	−0.715780	−0.357890	+	0.933764i	$0.616504\pi$
$104$	621.969	0.586434
$105$	0	0
$106$	−1197.82	−1.09757
$107$	1435.00	1.29651	0.648255	−	0.761423i	$-0.275499\pi$
$107$	1435.00	1.29651	0.648255	+	0.761423i	$0.275499\pi$
$108$	0	0
$109$	−83.1305	−0.0730501	−0.0365250	−	0.999333i	$-0.511629\pi$
$109$	−83.1305	−0.0730501	−0.0365250	+	0.999333i	$0.511629\pi$
$110$	95.7691	0.0830112
$111$	0	0
$112$	−38.2769	−0.0322931
$113$	972.196	0.809349	0.404675	−	0.914461i	$-0.367385\pi$
$113$	972.196	0.809349	0.404675	+	0.914461i	$0.367385\pi$
$114$	0	0
$115$	89.0539	0.0722115
$116$	502.508	0.402213
$117$	0	0
$118$	−1340.55	−1.04583
$119$	−6.79970	−0.00523805
$120$	0	0
$121$	2217.49	1.66603
$122$	929.061	0.689453
$123$	0	0
$124$	252.246	0.182680
$125$	−200.442	−0.143425
$126$	0	0
$127$	−1316.13	−0.919588	−0.459794	−	0.888026i	$-0.652077\pi$
$127$	−1316.13	−0.919588	−0.459794	+	0.888026i	$0.652077\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	124.992	0.0843272
$131$	−1115.28	−0.743833	−0.371917	−	0.928266i	$-0.621299\pi$
$131$	−1115.28	−0.743833	−0.371917	+	0.928266i	$0.621299\pi$
$132$	0	0
$133$	283.046	0.184536
$134$	−1538.14	−0.991604
$135$	0	0
$136$	22.7386	0.0143369
$137$	617.143	0.384862	0.192431	−	0.981311i	$-0.438363\pi$
$137$	617.143	0.384862	0.192431	+	0.981311i	$0.438363\pi$
$138$	0	0
$139$	2156.57	1.31596	0.657978	−	0.753037i	$-0.271412\pi$
$139$	2156.57	1.31596	0.657978	+	0.753037i	$0.271412\pi$
$140$	−7.69219	−0.00464364
$141$	0	0
$142$	−1223.14	−0.722842
$143$	4631.28	2.70830
$144$	0	0
$145$	100.985	0.0578368
$146$	1847.66	1.04735
$147$	0	0
$148$	−909.569	−0.505177
$149$	3571.69	1.96379	0.981893	−	0.189437i	$-0.0606662\pi$
$149$	3571.69	1.96379	0.981893	+	0.189437i	$0.0606662\pi$
$150$	0	0
$151$	−2261.43	−1.21876	−0.609379	−	0.792879i	$-0.708581\pi$
$151$	−2261.43	−1.21876	−0.609379	+	0.792879i	$0.708581\pi$
$152$	−946.523	−0.505086
$153$	0	0
$154$	−285.015	−0.149138
$155$	50.6918	0.0262688
$156$	0	0
$157$	1829.25	0.929875	0.464937	−	0.885344i	$-0.346077\pi$
$157$	1829.25	0.929875	0.464937	+	0.885344i	$0.346077\pi$
$158$	79.6152	0.0400876
$159$	0	0
$160$	25.7231	0.0127099
$161$	−265.031	−0.129735
$162$	0	0
$163$	268.554	0.129048	0.0645238	−	0.997916i	$-0.479447\pi$
$163$	268.554	0.129048	0.0645238	+	0.997916i	$0.479447\pi$
$164$	1298.52	0.618278
$165$	0	0
$166$	887.538	0.414978
$167$	−4054.52	−1.87873	−0.939366	−	0.342915i	$-0.888586\pi$
$167$	−4054.52	−1.87873	−0.939366	+	0.342915i	$0.888586\pi$
$168$	0	0
$169$	3847.46	1.75123
$170$	4.56959	0.00206160
$171$	0	0
$172$	−1091.29	−0.483781
$173$	1747.35	0.767911	0.383955	−	0.923352i	$-0.374562\pi$
$173$	1747.35	0.767911	0.383955	+	0.923352i	$0.374562\pi$
$174$	0	0
$175$	297.492	0.128505
$176$	953.108	0.408200
$177$	0	0
$178$	−156.915	−0.0660746
$179$	−2037.71	−0.850868	−0.425434	−	0.904989i	$-0.639879\pi$
$179$	−2037.71	−0.850868	−0.425434	+	0.904989i	$0.639879\pi$
$180$	0	0
$181$	2820.68	1.15834	0.579169	−	0.815207i	$-0.303377\pi$
$181$	2820.68	1.15834	0.579169	+	0.815207i	$0.303377\pi$
$182$	−371.985	−0.151502
$183$	0	0
$184$	886.277	0.355093
$185$	−182.789	−0.0726427
$186$	0	0
$187$	169.315	0.0662114
$188$	−19.7541	−0.00766336
$189$	0	0
$190$	−190.215	−0.0726297
$191$	439.723	0.166582	0.0832912	−	0.996525i	$-0.473457\pi$
$191$	439.723	0.166582	0.0832912	+	0.996525i	$0.473457\pi$
$192$	0	0
$193$	−462.676	−0.172560	−0.0862802	−	0.996271i	$-0.527498\pi$
$193$	−462.676	−0.172560	−0.0862802	+	0.996271i	$0.527498\pi$
$194$	−182.031	−0.0673663
$195$	0	0
$196$	−1349.11	−0.491657
$197$	1036.60	0.374896	0.187448	−	0.982275i	$-0.439978\pi$
$197$	1036.60	0.374896	0.187448	+	0.982275i	$0.439978\pi$
$198$	0	0
$199$	−1190.22	−0.423980	−0.211990	−	0.977272i	$-0.567994\pi$
$199$	−1190.22	−0.423980	−0.211990	+	0.977272i	$0.567994\pi$
$200$	−994.831	−0.351726
$201$	0	0
$202$	−3245.05	−1.13030
$203$	−300.538	−0.103909
$204$	0	0
$205$	260.954	0.0889063
$206$	−1496.46	−0.506133
$207$	0	0
$208$	1243.94	0.414671
$209$	−7047.95	−2.33262
$210$	0	0
$211$	−4928.01	−1.60786	−0.803929	−	0.594725i	$-0.797261\pi$
$211$	−4928.01	−1.60786	−0.803929	+	0.594725i	$0.797261\pi$
$212$	−2395.63	−0.776097
$213$	0	0
$214$	2870.00	0.916772
$215$	−219.308	−0.0695660
$216$	0	0
$217$	−150.862	−0.0471945
$218$	−166.261	−0.0516542
$219$	0	0
$220$	191.538	0.0586978
$221$	220.980	0.0672611
$222$	0	0
$223$	−1470.47	−0.441569	−0.220785	−	0.975323i	$-0.570862\pi$
$223$	−1470.47	−0.441569	−0.220785	+	0.975323i	$0.570862\pi$
$224$	−76.5538	−0.0228347
$225$	0	0
$226$	1944.39	0.572296
$227$	2102.26	0.614678	0.307339	−	0.951600i	$-0.400561\pi$
$227$	2102.26	0.614678	0.307339	+	0.951600i	$0.400561\pi$
$228$	0	0
$229$	−1001.04	−0.288867	−0.144433	−	0.989515i	$-0.546136\pi$
$229$	−1001.04	−0.288867	−0.144433	+	0.989515i	$0.546136\pi$
$230$	178.108	0.0510612
$231$	0	0
$232$	1005.02	0.284407
$233$	−6280.78	−1.76596	−0.882978	−	0.469415i	$-0.844465\pi$
$233$	−6280.78	−1.76596	−0.882978	+	0.469415i	$0.844465\pi$
$234$	0	0
$235$	−3.96981	−0.00110197
$236$	−2681.11	−0.739514
$237$	0	0
$238$	−13.5994	−0.00370386
$239$	−1017.37	−0.275348	−0.137674	−	0.990478i	$-0.543963\pi$
$239$	−1017.37	−0.275348	−0.137674	+	0.990478i	$0.543963\pi$
$240$	0	0
$241$	−921.292	−0.246247	−0.123124	−	0.992391i	$-0.539291\pi$
$241$	−921.292	−0.246247	−0.123124	+	0.992391i	$0.539291\pi$
$242$	4434.98	1.17806
$243$	0	0
$244$	1858.12	0.487517
$245$	−271.119	−0.0706987
$246$	0	0
$247$	−9198.56	−2.36960
$248$	504.492	0.129174
$249$	0	0
$250$	−400.885	−0.101417
$251$	2262.40	0.568930	0.284465	−	0.958686i	$-0.408184\pi$
$251$	2262.40	0.568930	0.284465	+	0.958686i	$0.408184\pi$
$252$	0	0
$253$	6599.35	1.63991
$254$	−2632.26	−0.650247
$255$	0	0
$256$	256.000	0.0625000
$257$	4786.65	1.16180	0.580901	−	0.813974i	$-0.302700\pi$
$257$	4786.65	1.16180	0.580901	+	0.813974i	$0.302700\pi$
$258$	0	0
$259$	543.992	0.130510
$260$	249.984	0.0596283
$261$	0	0
$262$	−2230.55	−0.525970
$263$	−3007.92	−0.705234	−0.352617	−	0.935768i	$-0.614708\pi$
$263$	−3007.92	−0.705234	−0.352617	+	0.935768i	$0.614708\pi$
$264$	0	0
$265$	−481.430	−0.111600
$266$	566.093	0.130486
$267$	0	0
$268$	−3076.28	−0.701170
$269$	−6559.73	−1.48682	−0.743409	−	0.668837i	$-0.766792\pi$
$269$	−6559.73	−1.48682	−0.743409	+	0.668837i	$0.766792\pi$
$270$	0	0
$271$	−5147.64	−1.15386	−0.576931	−	0.816793i	$-0.695750\pi$
$271$	−5147.64	−1.15386	−0.576931	+	0.816793i	$0.695750\pi$
$272$	45.4772	0.0101377
$273$	0	0
$274$	1234.29	0.272138
$275$	−7407.66	−1.62436
$276$	0	0
$277$	−1067.66	−0.231587	−0.115793	−	0.993273i	$-0.536941\pi$
$277$	−1067.66	−0.231587	−0.115793	+	0.993273i	$0.536941\pi$
$278$	4313.14	0.930521
$279$	0	0
$280$	−15.3844	−0.00328355
$281$	4186.47	0.888769	0.444384	−	0.895836i	$-0.353422\pi$
$281$	4186.47	0.888769	0.444384	+	0.895836i	$0.353422\pi$
$282$	0	0
$283$	5356.95	1.12522	0.562611	−	0.826722i	$-0.309797\pi$
$283$	5356.95	1.12522	0.562611	+	0.826722i	$0.309797\pi$
$284$	−2446.28	−0.511126
$285$	0	0
$286$	9262.55	1.91506
$287$	−776.616	−0.159729
$288$	0	0
$289$	−4904.92	−0.998356
$290$	201.970	0.0408968
$291$	0	0
$292$	3695.32	0.740590
$293$	2895.79	0.577385	0.288692	−	0.957422i	$-0.406780\pi$
$293$	2895.79	0.577385	0.288692	+	0.957422i	$0.406780\pi$
$294$	0	0
$295$	−538.800	−0.106340
$296$	−1819.14	−0.357214
$297$	0	0
$298$	7143.38	1.38861
$299$	8613.08	1.66591
$300$	0	0
$301$	652.676	0.124982
$302$	−4522.86	−0.861793
$303$	0	0
$304$	−1893.05	−0.357150
$305$	373.412	0.0701033
$306$	0	0
$307$	1855.49	0.344947	0.172473	−	0.985014i	$-0.444824\pi$
$307$	1855.49	0.344947	0.172473	+	0.985014i	$0.444824\pi$
$308$	−570.031	−0.105456
$309$	0	0
$310$	101.384	0.0185749
$311$	7171.69	1.30762	0.653809	−	0.756659i	$-0.273170\pi$
$311$	7171.69	1.30762	0.653809	+	0.756659i	$0.273170\pi$
$312$	0	0
$313$	−1310.83	−0.236717	−0.118359	−	0.992971i	$-0.537763\pi$
$313$	−1310.83	−0.236717	−0.118359	+	0.992971i	$0.537763\pi$
$314$	3658.51	0.657521
$315$	0	0
$316$	159.230	0.0283462
$317$	3689.26	0.653658	0.326829	−	0.945083i	$-0.394020\pi$
$317$	3689.26	0.653658	0.326829	+	0.945083i	$0.394020\pi$
$318$	0	0
$319$	7483.50	1.31347
$320$	51.4462	0.00898729
$321$	0	0
$322$	−530.061	−0.0917365
$323$	−336.290	−0.0579310
$324$	0	0
$325$	−9668.03	−1.65011
$326$	537.108	0.0912504
$327$	0	0
$328$	2597.05	0.437189
$329$	11.8144	0.00197979
$330$	0	0
$331$	−3244.96	−0.538849	−0.269425	−	0.963021i	$-0.586834\pi$
$331$	−3244.96	−0.538849	−0.269425	+	0.963021i	$0.586834\pi$
$332$	1775.08	0.293434
$333$	0	0
$334$	−8109.04	−1.32846
$335$	−618.214	−0.100826
$336$	0	0
$337$	4301.66	0.695331	0.347665	−	0.937619i	$-0.386975\pi$
$337$	4301.66	0.695331	0.347665	+	0.937619i	$0.386975\pi$
$338$	7694.92	1.23831
$339$	0	0
$340$	9.13918	0.00145777
$341$	3756.52	0.596561
$342$	0	0
$343$	1627.43	0.256189
$344$	−2182.58	−0.342085
$345$	0	0
$346$	3494.70	0.542995
$347$	−3304.20	−0.511178	−0.255589	−	0.966786i	$-0.582269\pi$
$347$	−3304.20	−0.511178	−0.255589	+	0.966786i	$0.582269\pi$
$348$	0	0
$349$	4508.71	0.691535	0.345767	−	0.938320i	$-0.387619\pi$
$349$	4508.71	0.691535	0.345767	+	0.938320i	$0.387619\pi$
$350$	594.985	0.0908665
$351$	0	0
$352$	1906.22	0.288641
$353$	6886.74	1.03837	0.519184	−	0.854662i	$-0.326236\pi$
$353$	6886.74	1.03837	0.519184	+	0.854662i	$0.326236\pi$
$354$	0	0
$355$	−491.608	−0.0734982
$356$	−313.830	−0.0467218
$357$	0	0
$358$	−4075.42	−0.601655
$359$	−1529.31	−0.224829	−0.112415	−	0.993661i	$-0.535859\pi$
$359$	−1529.31	−0.224829	−0.112415	+	0.993661i	$0.535859\pi$
$360$	0	0
$361$	7139.52	1.04090
$362$	5641.35	0.819069
$363$	0	0
$364$	−743.970	−0.107128
$365$	742.619	0.106494
$366$	0	0
$367$	−10231.4	−1.45524	−0.727620	−	0.685980i	$-0.759374\pi$
$367$	−10231.4	−1.45524	−0.727620	+	0.685980i	$0.759374\pi$
$368$	1772.55	0.251089
$369$	0	0
$370$	−365.578	−0.0513661
$371$	1432.77	0.200501
$372$	0	0
$373$	946.430	0.131379	0.0656894	−	0.997840i	$-0.479075\pi$
$373$	946.430	0.131379	0.0656894	+	0.997840i	$0.479075\pi$
$374$	338.630	0.0468186
$375$	0	0
$376$	−39.5081	−0.00541882
$377$	9767.01	1.33429
$378$	0	0
$379$	−8717.46	−1.18149	−0.590746	−	0.806857i	$-0.701167\pi$
$379$	−8717.46	−1.18149	−0.590746	+	0.806857i	$0.701167\pi$
$380$	−380.430	−0.0513570
$381$	0	0
$382$	879.446	0.117792
$383$	12261.5	1.63586	0.817931	−	0.575316i	$-0.195121\pi$
$383$	12261.5	1.63586	0.817931	+	0.575316i	$0.195121\pi$
$384$	0	0
$385$	−114.554	−0.0151643
$386$	−925.353	−0.122019
$387$	0	0
$388$	−364.062	−0.0476352
$389$	−13211.1	−1.72192	−0.860961	−	0.508672i	$-0.830137\pi$
$389$	−13211.1	−1.72192	−0.860961	+	0.508672i	$0.830137\pi$
$390$	0	0
$391$	314.886	0.0407275
$392$	−2698.22	−0.347654
$393$	0	0
$394$	2073.19	0.265091
$395$	31.9993	0.00407609
$396$	0	0
$397$	−5211.93	−0.658890	−0.329445	−	0.944175i	$-0.606862\pi$
$397$	−5211.93	−0.658890	−0.329445	+	0.944175i	$0.606862\pi$
$398$	−2380.43	−0.299799
$399$	0	0
$400$	−1989.66	−0.248708
$401$	985.634	0.122744	0.0613719	−	0.998115i	$-0.480452\pi$
$401$	985.634	0.122744	0.0613719	+	0.998115i	$0.480452\pi$
$402$	0	0
$403$	4902.79	0.606018
$404$	−6490.09	−0.799243
$405$	0	0
$406$	−601.076	−0.0734751
$407$	−13545.6	−1.64970
$408$	0	0
$409$	−674.647	−0.0815627	−0.0407813	−	0.999168i	$-0.512985\pi$
$409$	−674.647	−0.0815627	−0.0407813	+	0.999168i	$0.512985\pi$
$410$	521.907	0.0628662
$411$	0	0
$412$	−2992.92	−0.357890
$413$	1603.51	0.191049
$414$	0	0
$415$	356.723	0.0421948
$416$	2487.88	0.293217
$417$	0	0
$418$	−14095.9	−1.64941
$419$	−7505.63	−0.875117	−0.437559	−	0.899190i	$-0.644157\pi$
$419$	−7505.63	−0.875117	−0.437559	+	0.899190i	$0.644157\pi$
$420$	0	0
$421$	−1352.62	−0.156586	−0.0782932	−	0.996930i	$-0.524947\pi$
$421$	−1352.62	−0.156586	−0.0782932	+	0.996930i	$0.524947\pi$
$422$	−9856.01	−1.13693
$423$	0	0
$424$	−4791.26	−0.548783
$425$	−353.454	−0.0403412
$426$	0	0
$427$	−1111.30	−0.125947
$428$	5740.00	0.648255
$429$	0	0
$430$	−438.616	−0.0491906
$431$	4003.21	0.447397	0.223698	−	0.974658i	$-0.428187\pi$
$431$	4003.21	0.447397	0.223698	+	0.974658i	$0.428187\pi$
$432$	0	0
$433$	9975.38	1.10713	0.553564	−	0.832807i	$-0.313267\pi$
$433$	9975.38	1.10713	0.553564	+	0.832807i	$0.313267\pi$
$434$	−301.725	−0.0333715
$435$	0	0
$436$	−332.522	−0.0365250
$437$	−13107.5	−1.43482
$438$	0	0
$439$	17360.9	1.88745	0.943726	−	0.330728i	$-0.107294\pi$
$439$	17360.9	1.88745	0.943726	+	0.330728i	$0.107294\pi$
$440$	383.077	0.0415056
$441$	0	0
$442$	441.959	0.0475608
$443$	8575.43	0.919709	0.459854	−	0.887994i	$-0.347902\pi$
$443$	8575.43	0.919709	0.459854	+	0.887994i	$0.347902\pi$
$444$	0	0
$445$	−63.0679	−0.00671844
$446$	−2940.94	−0.312236
$447$	0	0
$448$	−153.108	−0.0161465
$449$	−4703.77	−0.494398	−0.247199	−	0.968965i	$-0.579510\pi$
$449$	−4703.77	−0.494398	−0.247199	+	0.968965i	$0.579510\pi$
$450$	0	0
$451$	19338.0	2.01905
$452$	3888.78	0.404675
$453$	0	0
$454$	4204.52	0.434643
$455$	−149.510	−0.0154047
$456$	0	0
$457$	−7046.80	−0.721303	−0.360651	−	0.932701i	$-0.617446\pi$
$457$	−7046.80	−0.721303	−0.360651	+	0.932701i	$0.617446\pi$
$458$	−2002.08	−0.204259
$459$	0	0
$460$	356.216	0.0361058
$461$	−5770.26	−0.582967	−0.291484	−	0.956576i	$-0.594149\pi$
$461$	−5770.26	−0.582967	−0.291484	+	0.956576i	$0.594149\pi$
$462$	0	0
$463$	−13599.1	−1.36502	−0.682511	−	0.730875i	$-0.739112\pi$
$463$	−13599.1	−1.36502	−0.682511	+	0.730875i	$0.739112\pi$
$464$	2010.03	0.201106
$465$	0	0
$466$	−12561.6	−1.24872
$467$	−5730.52	−0.567831	−0.283915	−	0.958849i	$-0.591633\pi$
$467$	−5730.52	−0.567831	−0.283915	+	0.958849i	$0.591633\pi$
$468$	0	0
$469$	1839.85	0.181143
$470$	−7.93962	−0.000779207 0
$471$	0	0
$472$	−5362.22	−0.522915
$473$	−16251.9	−1.57983
$474$	0	0
$475$	14713.0	1.42122
$476$	−27.1988	−0.00261902
$477$	0	0
$478$	−2034.74	−0.194701
$479$	−14568.3	−1.38965	−0.694824	−	0.719180i	$-0.744518\pi$
$479$	−14568.3	−1.38965	−0.694824	+	0.719180i	$0.744518\pi$
$480$	0	0
$481$	−17678.9	−1.67586
$482$	−1842.58	−0.174123
$483$	0	0
$484$	8869.97	0.833017
$485$	−73.1626	−0.00684977
$486$	0	0
$487$	7164.51	0.666642	0.333321	−	0.942813i	$-0.391831\pi$
$487$	7164.51	0.666642	0.333321	+	0.942813i	$0.391831\pi$
$488$	3716.25	0.344727
$489$	0	0
$490$	−542.238	−0.0499915
$491$	9510.06	0.874100	0.437050	−	0.899437i	$-0.356023\pi$
$491$	9510.06	0.874100	0.437050	+	0.899437i	$0.356023\pi$
$492$	0	0
$493$	357.072	0.0326201
$494$	−18397.1	−1.67556
$495$	0	0
$496$	1008.98	0.0913401
$497$	1463.06	0.132047
$498$	0	0
$499$	21485.6	1.92751	0.963756	−	0.266785i	$-0.0859614\pi$
$499$	21485.6	1.92751	0.963756	+	0.266785i	$0.0859614\pi$
$500$	−801.770	−0.0717125
$501$	0	0
$502$	4524.80	0.402294
$503$	11603.2	1.02855	0.514274	−	0.857626i	$-0.328061\pi$
$503$	11603.2	1.02855	0.514274	+	0.857626i	$0.328061\pi$
$504$	0	0
$505$	−1304.26	−0.114928
$506$	13198.7	1.15959
$507$	0	0
$508$	−5264.52	−0.459794
$509$	−8939.74	−0.778481	−0.389241	−	0.921136i	$-0.627262\pi$
$509$	−8939.74	−0.778481	−0.389241	+	0.921136i	$0.627262\pi$
$510$	0	0
$511$	−2210.08	−0.191328
$512$	512.000	0.0441942
$513$	0	0
$514$	9573.30	0.821518
$515$	−601.464	−0.0514634
$516$	0	0
$517$	−294.183	−0.0250255
$518$	1087.98	0.0922843
$519$	0	0
$520$	499.968	0.0421636
$521$	7410.10	0.623114	0.311557	−	0.950227i	$-0.399149\pi$
$521$	7410.10	0.623114	0.311557	+	0.950227i	$0.399149\pi$
$522$	0	0
$523$	18970.8	1.58611	0.793054	−	0.609151i	$-0.208490\pi$
$523$	18970.8	1.58611	0.793054	+	0.609151i	$0.208490\pi$
$524$	−4461.11	−0.371917
$525$	0	0
$526$	−6015.85	−0.498676
$527$	179.241	0.0148157
$528$	0	0
$529$	106.230	0.00873097
$530$	−962.861	−0.0789132
$531$	0	0
$532$	1132.19	0.0922678
$533$	25238.8	2.05106
$534$	0	0
$535$	1153.52	0.0932169
$536$	−6152.55	−0.495802
$537$	0	0
$538$	−13119.5	−1.05134
$539$	−20091.3	−1.60556
$540$	0	0
$541$	−11440.4	−0.909169	−0.454584	−	0.890704i	$-0.650212\pi$
$541$	−11440.4	−0.909169	−0.454584	+	0.890704i	$0.650212\pi$
$542$	−10295.3	−0.815904
$543$	0	0
$544$	90.9543	0.00716845
$545$	−66.8243	−0.00525218
$546$	0	0
$547$	10253.0	0.801434	0.400717	−	0.916202i	$-0.368761\pi$
$547$	10253.0	0.801434	0.400717	+	0.916202i	$0.368761\pi$
$548$	2468.57	0.192431
$549$	0	0
$550$	−14815.3	−1.14860
$551$	−14863.6	−1.14920
$552$	0	0
$553$	−95.2320	−0.00732310
$554$	−2135.32	−0.163757
$555$	0	0
$556$	8626.28	0.657978
$557$	15336.5	1.16666	0.583328	−	0.812236i	$-0.301750\pi$
$557$	15336.5	1.16666	0.583328	+	0.812236i	$0.301750\pi$
$558$	0	0
$559$	−21210.9	−1.60488
$560$	−30.7688	−0.00232182
$561$	0	0
$562$	8372.95	0.628454
$563$	7701.37	0.576508	0.288254	−	0.957554i	$-0.406925\pi$
$563$	7701.37	0.576508	0.288254	+	0.957554i	$0.406925\pi$
$564$	0	0
$565$	781.497	0.0581909
$566$	10713.9	0.795652
$567$	0	0
$568$	−4892.55	−0.361421
$569$	−17739.3	−1.30697	−0.653487	−	0.756938i	$-0.726695\pi$
$569$	−17739.3	−1.30697	−0.653487	+	0.756938i	$0.726695\pi$
$570$	0	0
$571$	13823.1	1.01310	0.506549	−	0.862211i	$-0.330921\pi$
$571$	13823.1	1.01310	0.506549	+	0.862211i	$0.330921\pi$
$572$	18525.1	1.35415
$573$	0	0
$574$	−1553.23	−0.112945
$575$	−13776.5	−0.999164
$576$	0	0
$577$	−15339.1	−1.10672	−0.553359	−	0.832943i	$-0.686654\pi$
$577$	−15339.1	−1.10672	−0.553359	+	0.832943i	$0.686654\pi$
$578$	−9809.84	−0.705944
$579$	0	0
$580$	403.940	0.0289184
$581$	−1061.63	−0.0758070
$582$	0	0
$583$	−35676.5	−2.53442
$584$	7390.65	0.523676
$585$	0	0
$586$	5791.58	0.408273
$587$	14766.5	1.03830	0.519148	−	0.854684i	$-0.326249\pi$
$587$	14766.5	1.03830	0.519148	+	0.854684i	$0.326249\pi$
$588$	0	0
$589$	−7461.14	−0.521954
$590$	−1077.60	−0.0751934
$591$	0	0
$592$	−3638.28	−0.252588
$593$	17603.6	1.21904	0.609522	−	0.792769i	$-0.291361\pi$
$593$	17603.6	1.21904	0.609522	+	0.792769i	$0.291361\pi$
$594$	0	0
$595$	−5.46593	−0.000376607 0
$596$	14286.8	0.981893
$597$	0	0
$598$	17226.2	1.17798
$599$	22801.1	1.55531	0.777653	−	0.628694i	$-0.216410\pi$
$599$	22801.1	1.55531	0.777653	+	0.628694i	$0.216410\pi$
$600$	0	0
$601$	1905.25	0.129312	0.0646562	−	0.997908i	$-0.479405\pi$
$601$	1905.25	0.129312	0.0646562	+	0.997908i	$0.479405\pi$
$602$	1305.35	0.0883757
$603$	0	0
$604$	−9045.72	−0.609379
$605$	1782.53	0.119785
$606$	0	0
$607$	−2589.22	−0.173136	−0.0865678	−	0.996246i	$-0.527590\pi$
$607$	−2589.22	−0.173136	−0.0865678	+	0.996246i	$0.527590\pi$
$608$	−3786.09	−0.252543
$609$	0	0
$610$	746.824	0.0495705
$611$	−383.950	−0.0254222
$612$	0	0
$613$	16930.4	1.11552	0.557759	−	0.830003i	$-0.311661\pi$
$613$	16930.4	1.11552	0.557759	+	0.830003i	$0.311661\pi$
$614$	3710.99	0.243914
$615$	0	0
$616$	−1140.06	−0.0745688
$617$	12813.4	0.836061	0.418030	−	0.908433i	$-0.362721\pi$
$617$	12813.4	0.836061	0.418030	+	0.908433i	$0.362721\pi$
$618$	0	0
$619$	10267.6	0.666703	0.333351	−	0.942803i	$-0.391820\pi$
$619$	10267.6	0.666703	0.333351	+	0.942803i	$0.391820\pi$
$620$	202.767	0.0131344
$621$	0	0
$622$	14343.4	0.924626
$623$	187.694	0.0120703
$624$	0	0
$625$	15383.1	0.984519
$626$	−2621.66	−0.167384
$627$	0	0
$628$	7317.01	0.464937
$629$	−646.322	−0.0409707
$630$	0	0
$631$	−21887.5	−1.38087	−0.690433	−	0.723397i	$-0.742580\pi$
$631$	−21887.5	−1.38087	−0.690433	+	0.723397i	$0.742580\pi$
$632$	318.461	0.0200438
$633$	0	0
$634$	7378.53	0.462206
$635$	−1057.97	−0.0661168
$636$	0	0
$637$	−26222.0	−1.63101
$638$	14967.0	0.928760
$639$	0	0
$640$	102.892	0.00635497
$641$	17203.7	1.06007	0.530035	−	0.847976i	$-0.322179\pi$
$641$	17203.7	1.06007	0.530035	+	0.847976i	$0.322179\pi$
$642$	0	0
$643$	12565.6	0.770666	0.385333	−	0.922778i	$-0.374087\pi$
$643$	12565.6	0.770666	0.385333	+	0.922778i	$0.374087\pi$
$644$	−1060.12	−0.0648675
$645$	0	0
$646$	−672.581	−0.0409634
$647$	3552.55	0.215866	0.107933	−	0.994158i	$-0.465577\pi$
$647$	3552.55	0.215866	0.107933	+	0.994158i	$0.465577\pi$
$648$	0	0
$649$	−39927.9	−2.41496
$650$	−19336.1	−1.16680
$651$	0	0
$652$	1074.22	0.0645238
$653$	5771.00	0.345845	0.172922	−	0.984935i	$-0.444679\pi$
$653$	5771.00	0.345845	0.172922	+	0.984935i	$0.444679\pi$
$654$	0	0
$655$	−896.512	−0.0534804
$656$	5194.09	0.309139
$657$	0	0
$658$	23.6289	0.00139992
$659$	16478.4	0.974063	0.487032	−	0.873384i	$-0.338080\pi$
$659$	16478.4	0.974063	0.487032	+	0.873384i	$0.338080\pi$
$660$	0	0
$661$	−4563.24	−0.268516	−0.134258	−	0.990946i	$-0.542865\pi$
$661$	−4563.24	−0.268516	−0.134258	+	0.990946i	$0.542865\pi$
$662$	−6489.92	−0.381024
$663$	0	0
$664$	3550.15	0.207489
$665$	227.526	0.0132678
$666$	0	0
$667$	13917.5	0.807929
$668$	−16218.1	−0.939366
$669$	0	0
$670$	−1236.43	−0.0712946
$671$	27671.7	1.59203
$672$	0	0
$673$	−13152.3	−0.753318	−0.376659	−	0.926352i	$-0.622927\pi$
$673$	−13152.3	−0.753318	−0.376659	+	0.926352i	$0.622927\pi$
$674$	8603.32	0.491673
$675$	0	0
$676$	15389.8	0.875617
$677$	14876.1	0.844515	0.422257	−	0.906476i	$-0.361238\pi$
$677$	14876.1	0.844515	0.422257	+	0.906476i	$0.361238\pi$
$678$	0	0
$679$	217.737	0.0123063
$680$	18.2784	0.00103080
$681$	0	0
$682$	7513.05	0.421832
$683$	−24082.4	−1.34918	−0.674588	−	0.738195i	$-0.735679\pi$
$683$	−24082.4	−1.34918	−0.674588	+	0.738195i	$0.735679\pi$
$684$	0	0
$685$	496.089	0.0276709
$686$	3254.86	0.181153
$687$	0	0
$688$	−4365.17	−0.241890
$689$	−46562.7	−2.57460
$690$	0	0
$691$	12890.2	0.709648	0.354824	−	0.934933i	$-0.384541\pi$
$691$	12890.2	0.709648	0.354824	+	0.934933i	$0.384541\pi$
$692$	6989.40	0.383955
$693$	0	0
$694$	−6608.40	−0.361457
$695$	1733.55	0.0946150
$696$	0	0
$697$	922.705	0.0501434
$698$	9017.41	0.488989
$699$	0	0
$700$	1189.97	0.0642523
$701$	−11887.0	−0.640467	−0.320233	−	0.947339i	$-0.603761\pi$
$701$	−11887.0	−0.640467	−0.320233	+	0.947339i	$0.603761\pi$
$702$	0	0
$703$	26904.0	1.44339
$704$	3812.43	0.204100
$705$	0	0
$706$	13773.5	0.734237
$707$	3881.57	0.206480
$708$	0	0
$709$	5814.43	0.307991	0.153996	−	0.988072i	$-0.450786\pi$
$709$	5814.43	0.307991	0.153996	+	0.988072i	$0.450786\pi$
$710$	−983.217	−0.0519711
$711$	0	0
$712$	−627.661	−0.0330373
$713$	6986.24	0.366952
$714$	0	0
$715$	3722.84	0.194722
$716$	−8150.83	−0.425434
$717$	0	0
$718$	−3058.61	−0.158978
$719$	4488.23	0.232800	0.116400	−	0.993202i	$-0.462865\pi$
$719$	4488.23	0.232800	0.116400	+	0.993202i	$0.462865\pi$
$720$	0	0
$721$	1790.00	0.0924590
$722$	14279.0	0.736026
$723$	0	0
$724$	11282.7	0.579169
$725$	−15622.2	−0.800267
$726$	0	0
$727$	23184.0	1.18273	0.591366	−	0.806403i	$-0.298589\pi$
$727$	23184.0	1.18273	0.591366	+	0.806403i	$0.298589\pi$
$728$	−1487.94	−0.0757510
$729$	0	0
$730$	1485.24	0.0753029
$731$	−775.451	−0.0392354
$732$	0	0
$733$	−20677.4	−1.04193	−0.520967	−	0.853577i	$-0.674429\pi$
$733$	−20677.4	−1.04193	−0.520967	+	0.853577i	$0.674429\pi$
$734$	−20462.7	−1.02901
$735$	0	0
$736$	3545.11	0.177547
$737$	−45812.8	−2.28974
$738$	0	0
$739$	−13001.7	−0.647191	−0.323595	−	0.946196i	$-0.604892\pi$
$739$	−13001.7	−0.647191	−0.323595	+	0.946196i	$0.604892\pi$
$740$	−731.155	−0.0363213
$741$	0	0
$742$	2865.54	0.141775
$743$	9267.09	0.457573	0.228786	−	0.973477i	$-0.426524\pi$
$743$	9267.09	0.457573	0.228786	+	0.973477i	$0.426524\pi$
$744$	0	0
$745$	2871.09	0.141193
$746$	1892.86	0.0928988
$747$	0	0
$748$	677.260	0.0331057
$749$	−3432.96	−0.167473
$750$	0	0
$751$	−30722.3	−1.49277	−0.746385	−	0.665514i	$-0.768212\pi$
$751$	−30722.3	−1.49277	−0.746385	+	0.665514i	$0.768212\pi$
$752$	−79.0162	−0.00383168
$753$	0	0
$754$	19534.0	0.943484
$755$	−1817.85	−0.0876267
$756$	0	0
$757$	5356.86	0.257197	0.128599	−	0.991697i	$-0.458952\pi$
$757$	5356.86	0.257197	0.128599	+	0.991697i	$0.458952\pi$
$758$	−17434.9	−0.835442
$759$	0	0
$760$	−760.860	−0.0363149
$761$	10019.0	0.477252	0.238626	−	0.971111i	$-0.423303\pi$
$761$	10019.0	0.477252	0.238626	+	0.971111i	$0.423303\pi$
$762$	0	0
$763$	198.874	0.00943605
$764$	1758.89	0.0832912
$765$	0	0
$766$	24523.1	1.15673
$767$	−52111.4	−2.45324
$768$	0	0
$769$	−17551.1	−0.823029	−0.411514	−	0.911403i	$-0.635000\pi$
$769$	−17551.1	−0.823029	−0.411514	+	0.911403i	$0.635000\pi$
$770$	−229.109	−0.0107227
$771$	0	0
$772$	−1850.71	−0.0862802
$773$	17144.5	0.797731	0.398866	−	0.917009i	$-0.369404\pi$
$773$	17144.5	0.797731	0.398866	+	0.917009i	$0.369404\pi$
$774$	0	0
$775$	−7841.94	−0.363472
$776$	−728.124	−0.0336831
$777$	0	0
$778$	−26422.1	−1.21758
$779$	−38408.8	−1.76654
$780$	0	0
$781$	−36430.7	−1.66913
$782$	629.771	0.0287987
$783$	0	0
$784$	−5396.43	−0.245829
$785$	1470.44	0.0668564
$786$	0	0
$787$	−8595.46	−0.389320	−0.194660	−	0.980871i	$-0.562360\pi$
$787$	−8595.46	−0.389320	−0.194660	+	0.980871i	$0.562360\pi$
$788$	4146.39	0.187448
$789$	0	0
$790$	63.9985	0.00288223
$791$	−2325.79	−0.104546
$792$	0	0
$793$	36115.5	1.61727
$794$	−10423.9	−0.465906
$795$	0	0
$796$	−4760.86	−0.211990
$797$	20837.8	0.926115	0.463057	−	0.886328i	$-0.346752\pi$
$797$	20837.8	0.926115	0.463057	+	0.886328i	$0.346752\pi$
$798$	0	0
$799$	−14.0369	−0.000621512 0
$800$	−3979.32	−0.175863
$801$	0	0
$802$	1971.27	0.0867929
$803$	55031.9	2.41847
$804$	0	0
$805$	−213.044	−0.00932773
$806$	9805.57	0.428519
$807$	0	0
$808$	−12980.2	−0.565150
$809$	6529.43	0.283761	0.141880	−	0.989884i	$-0.454685\pi$
$809$	6529.43	0.283761	0.141880	+	0.989884i	$0.454685\pi$
$810$	0	0
$811$	29764.0	1.28873	0.644363	−	0.764720i	$-0.277123\pi$
$811$	29764.0	1.28873	0.644363	+	0.764720i	$0.277123\pi$
$812$	−1202.15	−0.0519547
$813$	0	0
$814$	−27091.2	−1.16652
$815$	215.876	0.00927830
$816$	0	0
$817$	32279.2	1.38226
$818$	−1349.29	−0.0576735
$819$	0	0
$820$	1043.81	0.0444531
$821$	−29571.2	−1.25705	−0.628527	−	0.777788i	$-0.716342\pi$
$821$	−29571.2	−1.25705	−0.628527	+	0.777788i	$0.716342\pi$
$822$	0	0
$823$	6112.43	0.258889	0.129445	−	0.991587i	$-0.458681\pi$
$823$	6112.43	0.258889	0.129445	+	0.991587i	$0.458681\pi$
$824$	−5985.85	−0.253067
$825$	0	0
$826$	3207.01	0.135092
$827$	−3265.87	−0.137322	−0.0686612	−	0.997640i	$-0.521873\pi$
$827$	−3265.87	−0.137322	−0.0686612	+	0.997640i	$0.521873\pi$
$828$	0	0
$829$	−19327.8	−0.809747	−0.404874	−	0.914373i	$-0.632685\pi$
$829$	−19327.8	−0.809747	−0.404874	+	0.914373i	$0.632685\pi$
$830$	713.446	0.0298362
$831$	0	0
$832$	4975.75	0.207336
$833$	−958.650	−0.0398743
$834$	0	0
$835$	−3259.22	−0.135078
$836$	−28191.8	−1.16631
$837$	0	0
$838$	−15011.3	−0.618801
$839$	−37638.1	−1.54876	−0.774382	−	0.632719i	$-0.781939\pi$
$839$	−37638.1	−1.54876	−0.774382	+	0.632719i	$0.781939\pi$
$840$	0	0
$841$	−8606.87	−0.352900
$842$	−2705.25	−0.110723
$843$	0	0
$844$	−19712.0	−0.803929
$845$	3092.77	0.125911
$846$	0	0
$847$	−5304.92	−0.215206
$848$	−9582.52	−0.388049
$849$	0	0
$850$	−706.908	−0.0285256
$851$	−25191.6	−1.01475
$852$	0	0
$853$	41869.6	1.68064	0.840322	−	0.542088i	$-0.182366\pi$
$853$	41869.6	1.68064	0.840322	+	0.542088i	$0.182366\pi$
$854$	−2222.60	−0.0890583
$855$	0	0
$856$	11480.0	0.458386
$857$	−33072.9	−1.31826	−0.659131	−	0.752029i	$-0.729076\pi$
$857$	−33072.9	−1.31826	−0.659131	+	0.752029i	$0.729076\pi$
$858$	0	0
$859$	−14609.7	−0.580298	−0.290149	−	0.956981i	$-0.593705\pi$
$859$	−14609.7	−0.580298	−0.290149	+	0.956981i	$0.593705\pi$
$860$	−877.233	−0.0347830
$861$	0	0
$862$	8006.43	0.316357
$863$	−118.786	−0.00468543	−0.00234271	−	0.999997i	$-0.500746\pi$
$863$	−118.786	−0.00468543	−0.00234271	+	0.999997i	$0.500746\pi$
$864$	0	0
$865$	1404.60	0.0552115
$866$	19950.8	0.782857
$867$	0	0
$868$	−603.449	−0.0235972
$869$	2371.31	0.0925675
$870$	0	0
$871$	−59792.1	−2.32604
$872$	−665.044	−0.0258271
$873$	0	0
$874$	−26215.0	−1.01457
$875$	479.519	0.0185265
$876$	0	0
$877$	−6360.18	−0.244889	−0.122445	−	0.992475i	$-0.539073\pi$
$877$	−6360.18	−0.244889	−0.122445	+	0.992475i	$0.539073\pi$
$878$	34721.8	1.33463
$879$	0	0
$880$	766.153	0.0293489
$881$	32601.5	1.24673	0.623367	−	0.781930i	$-0.285764\pi$
$881$	32601.5	1.24673	0.623367	+	0.781930i	$0.285764\pi$
$882$	0	0
$883$	1478.26	0.0563392	0.0281696	−	0.999603i	$-0.491032\pi$
$883$	1478.26	0.0563392	0.0281696	+	0.999603i	$0.491032\pi$
$884$	883.919	0.0336305
$885$	0	0
$886$	17150.9	0.650332
$887$	45267.1	1.71355	0.856776	−	0.515688i	$-0.172464\pi$
$887$	45267.1	1.71355	0.856776	+	0.515688i	$0.172464\pi$
$888$	0	0
$889$	3148.59	0.118785
$890$	−126.136	−0.00475066
$891$	0	0
$892$	−5881.88	−0.220785
$893$	584.302	0.0218958
$894$	0	0
$895$	−1638.01	−0.0611760
$896$	−306.215	−0.0114173
$897$	0	0
$898$	−9407.54	−0.349592
$899$	7922.22	0.293905
$900$	0	0
$901$	−1702.29	−0.0629428
$902$	38676.0	1.42768
$903$	0	0
$904$	7777.57	0.286148
$905$	2267.39	0.0832826
$906$	0	0
$907$	7261.52	0.265838	0.132919	−	0.991127i	$-0.457565\pi$
$907$	7261.52	0.265838	0.132919	+	0.991127i	$0.457565\pi$
$908$	8409.04	0.307339
$909$	0	0
$910$	−299.019	−0.0108927
$911$	−19434.4	−0.706795	−0.353398	−	0.935473i	$-0.614974\pi$
$911$	−19434.4	−0.706795	−0.353398	+	0.935473i	$0.614974\pi$
$912$	0	0
$913$	26435.0	0.958237
$914$	−14093.6	−0.510038
$915$	0	0
$916$	−4004.15	−0.144433
$917$	2668.08	0.0960827
$918$	0	0
$919$	−24010.2	−0.861830	−0.430915	−	0.902392i	$-0.641809\pi$
$919$	−24010.2	−0.861830	−0.430915	+	0.902392i	$0.641809\pi$
$920$	712.432	0.0255306
$921$	0	0
$922$	−11540.5	−0.412220
$923$	−47547.1	−1.69559
$924$	0	0
$925$	28277.1	1.00513
$926$	−27198.3	−0.965217
$927$	0	0
$928$	4020.06	0.142204
$929$	41554.6	1.46756	0.733780	−	0.679387i	$-0.237754\pi$
$929$	41554.6	1.46756	0.733780	+	0.679387i	$0.237754\pi$
$930$	0	0
$931$	39905.0	1.40476
$932$	−25123.1	−0.882978
$933$	0	0
$934$	−11461.0	−0.401517
$935$	136.103	0.00476049
$936$	0	0
$937$	16811.0	0.586116	0.293058	−	0.956095i	$-0.405327\pi$
$937$	16811.0	0.586116	0.293058	+	0.956095i	$0.405327\pi$
$938$	3679.70	0.128088
$939$	0	0
$940$	−15.8792	−0.000550983 0
$941$	−10732.9	−0.371819	−0.185909	−	0.982567i	$-0.559523\pi$
$941$	−10732.9	−0.371819	−0.185909	+	0.982567i	$0.559523\pi$
$942$	0	0
$943$	35964.1	1.24194
$944$	−10724.4	−0.369757
$945$	0	0
$946$	−32503.7	−1.11711
$947$	31549.3	1.08259	0.541297	−	0.840832i	$-0.317934\pi$
$947$	31549.3	1.08259	0.541297	+	0.840832i	$0.317934\pi$
$948$	0	0
$949$	71824.3	2.45681
$950$	29425.9	1.00495
$951$	0	0
$952$	−54.3976	−0.00185193
$953$	31955.2	1.08618	0.543090	−	0.839675i	$-0.317254\pi$
$953$	31955.2	1.08618	0.543090	+	0.839675i	$0.317254\pi$
$954$	0	0
$955$	353.470	0.0119770
$956$	−4069.48	−0.137674
$957$	0	0
$958$	−29136.5	−0.982630
$959$	−1476.39	−0.0497135
$960$	0	0
$961$	−25814.2	−0.866512
$962$	−35357.7	−1.18501
$963$	0	0
$964$	−3685.17	−0.123124
$965$	−371.921	−0.0124068
$966$	0	0
$967$	38372.2	1.27608	0.638038	−	0.770005i	$-0.279746\pi$
$967$	38372.2	1.27608	0.638038	+	0.770005i	$0.279746\pi$
$968$	17739.9	0.589032
$969$	0	0
$970$	−146.325	−0.00484352
$971$	46445.9	1.53503	0.767517	−	0.641028i	$-0.221492\pi$
$971$	46445.9	1.53503	0.767517	+	0.641028i	$0.221492\pi$
$972$	0	0
$973$	−5159.17	−0.169985
$974$	14329.0	0.471387
$975$	0	0
$976$	7432.49	0.243758
$977$	47908.3	1.56880	0.784402	−	0.620253i	$-0.212970\pi$
$977$	47908.3	1.56880	0.784402	+	0.620253i	$0.212970\pi$
$978$	0	0
$979$	−4673.66	−0.152575
$980$	−1084.48	−0.0353493
$981$	0	0
$982$	19020.1	0.618082
$983$	40547.7	1.31564	0.657818	−	0.753177i	$-0.271480\pi$
$983$	40547.7	1.31564	0.657818	+	0.753177i	$0.271480\pi$
$984$	0	0
$985$	833.265	0.0269544
$986$	714.145	0.0230659
$987$	0	0
$988$	−36794.2	−1.18480
$989$	−30224.6	−0.971776
$990$	0	0
$991$	−10651.5	−0.341428	−0.170714	−	0.985321i	$-0.554607\pi$
$991$	−10651.5	−0.341428	−0.170714	+	0.985321i	$0.554607\pi$
$992$	2017.97	0.0645872
$993$	0	0
$994$	2926.12	0.0933711
$995$	−956.751	−0.0304835
$996$	0	0
$997$	−18758.4	−0.595871	−0.297935	−	0.954586i	$-0.596298\pi$
$997$	−18758.4	−0.595871	−0.297935	+	0.954586i	$0.596298\pi$
$998$	42971.2	1.36296
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.a.h.1.1	yes	2
3.2	odd	2		162.4.a.e.1.2	✓	2
4.3	odd	2		1296.4.a.s.1.1		2
9.2	odd	6		162.4.c.j.109.1		4
9.4	even	3		162.4.c.i.55.2		4
9.5	odd	6		162.4.c.j.55.1		4
9.7	even	3		162.4.c.i.109.2		4
12.11	even	2		1296.4.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.a.e.1.2	✓	2	3.2	odd	2
162.4.a.h.1.1	yes	2	1.1	even	1	trivial
162.4.c.i.55.2		4	9.4	even	3
162.4.c.i.109.2		4	9.7	even	3
162.4.c.j.55.1		4	9.5	odd	6
162.4.c.j.109.1		4	9.2	odd	6
1296.4.a.j.1.2		2	12.11	even	2
1296.4.a.s.1.1		2	4.3	odd	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +0.803848 q^{5} -2.39230 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +0.803848 q^{5} -2.39230 q^{7} +8.00000 q^{8} +1.60770 q^{10} +59.5692 q^{11} +77.7461 q^{13} -4.78461 q^{14} +16.0000 q^{16} +2.84232 q^{17} -118.315 q^{19} +3.21539 q^{20} +119.138 q^{22} +110.785 q^{23} -124.354 q^{25} +155.492 q^{26} -9.56922 q^{28} +125.627 q^{29} +63.0615 q^{31} +32.0000 q^{32} +5.68465 q^{34} -1.92305 q^{35} -227.392 q^{37} -236.631 q^{38} +6.43078 q^{40} +324.631 q^{41} -272.823 q^{43} +238.277 q^{44} +221.569 q^{46} -4.93851 q^{47} -337.277 q^{49} -248.708 q^{50} +310.985 q^{52} -598.908 q^{53} +47.8846 q^{55} -19.1384 q^{56} +251.254 q^{58} -670.277 q^{59} +464.531 q^{61} +126.123 q^{62} +64.0000 q^{64} +62.4960 q^{65} -769.069 q^{67} +11.3693 q^{68} -3.84610 q^{70} -611.569 q^{71} +923.831 q^{73} -454.785 q^{74} -473.261 q^{76} -142.508 q^{77} +39.8076 q^{79} +12.8616 q^{80} +649.261 q^{82} +443.769 q^{83} +2.28479 q^{85} -545.646 q^{86} +476.554 q^{88} -78.4576 q^{89} -185.992 q^{91} +443.138 q^{92} -9.87703 q^{94} -95.1075 q^{95} -91.0155 q^{97} -674.554 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{7} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^7 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{7} + 16 q^{8} + 24 q^{10} + 36 q^{11} + 10 q^{13} + 32 q^{14} + 32 q^{16} + 120 q^{17} - 8 q^{19} + 48 q^{20} + 72 q^{22} + 180 q^{23} - 124 q^{25} + 20 q^{26} + 64 q^{28} + 324 q^{29} - 248 q^{31} + 64 q^{32} + 240 q^{34} + 204 q^{35} - 434 q^{37} - 16 q^{38} + 96 q^{40} + 192 q^{41} - 608 q^{43} + 144 q^{44} + 360 q^{46} - 384 q^{47} - 342 q^{49} - 248 q^{50} + 40 q^{52} - 408 q^{53} - 216 q^{55} + 128 q^{56} + 648 q^{58} - 1008 q^{59} + 742 q^{61} - 496 q^{62} + 128 q^{64} - 696 q^{65} - 104 q^{67} + 480 q^{68} + 408 q^{70} - 1140 q^{71} + 850 q^{73} - 868 q^{74} - 32 q^{76} - 576 q^{77} - 440 q^{79} + 192 q^{80} + 384 q^{82} + 264 q^{83} + 1314 q^{85} - 1216 q^{86} + 288 q^{88} + 768 q^{89} - 1432 q^{91} + 720 q^{92} - 768 q^{94} + 1140 q^{95} - 764 q^{97} - 684 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^7 + 16 * q^8 + 24 * q^10 + 36 * q^11 + 10 * q^13 + 32 * q^14 + 32 * q^16 + 120 * q^17 - 8 * q^19 + 48 * q^20 + 72 * q^22 + 180 * q^23 - 124 * q^25 + 20 * q^26 + 64 * q^28 + 324 * q^29 - 248 * q^31 + 64 * q^32 + 240 * q^34 + 204 * q^35 - 434 * q^37 - 16 * q^38 + 96 * q^40 + 192 * q^41 - 608 * q^43 + 144 * q^44 + 360 * q^46 - 384 * q^47 - 342 * q^49 - 248 * q^50 + 40 * q^52 - 408 * q^53 - 216 * q^55 + 128 * q^56 + 648 * q^58 - 1008 * q^59 + 742 * q^61 - 496 * q^62 + 128 * q^64 - 696 * q^65 - 104 * q^67 + 480 * q^68 + 408 * q^70 - 1140 * q^71 + 850 * q^73 - 868 * q^74 - 32 * q^76 - 576 * q^77 - 440 * q^79 + 192 * q^80 + 384 * q^82 + 264 * q^83 + 1314 * q^85 - 1216 * q^86 + 288 * q^88 + 768 * q^89 - 1432 * q^91 + 720 * q^92 - 768 * q^94 + 1140 * q^95 - 764 * q^97 - 684 * q^98

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