LMFDB - Embedded newform 162.4.a.f.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(\sqrt{105}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 26 $ x^2 - x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 18)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.62348$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} -19.8704 q^{5} -5.87043 q^{7} -8.00000 q^{8} +39.7409 q^{10} +18.7409 q^{11} +45.8704 q^{13} +11.7409 q^{14} +16.0000 q^{16} +16.8704 q^{17} -10.3521 q^{19} -79.4817 q^{20} -37.4817 q^{22} +49.8704 q^{23} +269.834 q^{25} -91.7409 q^{26} -23.4817 q^{28} +10.9070 q^{29} +151.611 q^{31} -32.0000 q^{32} -33.7409 q^{34} +116.648 q^{35} +346.186 q^{37} +20.7043 q^{38} +158.963 q^{40} +264.741 q^{41} -411.890 q^{43} +74.9634 q^{44} -99.7409 q^{46} +472.056 q^{47} -308.538 q^{49} -539.668 q^{50} +183.482 q^{52} +290.186 q^{53} -372.389 q^{55} +46.9634 q^{56} -21.8140 q^{58} +53.2591 q^{59} -293.944 q^{61} -303.223 q^{62} +64.0000 q^{64} -911.465 q^{65} -398.555 q^{67} +67.4817 q^{68} -233.296 q^{70} -647.854 q^{71} -478.279 q^{73} -692.372 q^{74} -41.4085 q^{76} -110.017 q^{77} +374.316 q^{79} -317.927 q^{80} -529.482 q^{82} +933.279 q^{83} -335.223 q^{85} +823.780 q^{86} -149.927 q^{88} +368.817 q^{89} -269.279 q^{91} +199.482 q^{92} -944.113 q^{94} +205.701 q^{95} +274.149 q^{97} +617.076 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} - 9 q^{5} + 19 q^{7} - 16 q^{8} + 18 q^{10} - 24 q^{11} + 61 q^{13} - 38 q^{14} + 32 q^{16} + 3 q^{17} + 133 q^{19} - 36 q^{20} + 48 q^{22} + 69 q^{23} + 263 q^{25} - 122 q^{26}+ \cdots + 66 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 - 9 * q^5 + 19 * q^7 - 16 * q^8 + 18 * q^10 - 24 * q^11 + 61 * q^13 - 38 * q^14 + 32 * q^16 + 3 * q^17 + 133 * q^19 - 36 * q^20 + 48 * q^22 + 69 * q^23 + 263 * q^25 - 122 * q^26 + 76 * q^28 + 237 * q^29 + 211 * q^31 - 64 * q^32 - 6 * q^34 + 387 * q^35 + 262 * q^37 - 266 * q^38 + 72 * q^40 + 468 * q^41 - 86 * q^43 - 96 * q^44 - 138 * q^46 + 483 * q^47 - 33 * q^49 - 526 * q^50 + 244 * q^52 + 150 * q^53 - 837 * q^55 - 152 * q^56 - 474 * q^58 + 168 * q^59 - 1049 * q^61 - 422 * q^62 + 128 * q^64 - 747 * q^65 - 1166 * q^67 + 12 * q^68 - 774 * q^70 - 312 * q^71 - 311 * q^73 - 524 * q^74 + 532 * q^76 - 1173 * q^77 + 349 * q^79 - 144 * q^80 - 936 * q^82 + 1221 * q^83 - 486 * q^85 + 172 * q^86 + 192 * q^88 - 492 * q^89 + 107 * q^91 + 276 * q^92 - 966 * q^94 + 1764 * q^95 - 128 * q^97 + 66 * q^98

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
$5$	−19.8704	−1.77726	−0.888632	−	0.458620i	$-0.848344\pi$
$5$	−19.8704	−1.77726	−0.888632	+	0.458620i	$0.848344\pi$
$6$	0	0
$7$	−5.87043	−0.316973	−0.158487	−	0.987361i	$-0.550661\pi$
$7$	−5.87043	−0.316973	−0.158487	+	0.987361i	$0.550661\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	39.7409	1.25672
$11$	18.7409	0.513689	0.256845	−	0.966453i	$-0.417317\pi$
$11$	18.7409	0.513689	0.256845	+	0.966453i	$0.417317\pi$
$12$	0	0
$13$	45.8704	0.978628	0.489314	−	0.872108i	$-0.337247\pi$
$13$	45.8704	0.978628	0.489314	+	0.872108i	$0.337247\pi$
$14$	11.7409	0.224134
$15$	0	0
$16$	16.0000	0.250000
$17$	16.8704	0.240687	0.120344	−	0.992732i	$-0.461600\pi$
$17$	16.8704	0.240687	0.120344	+	0.992732i	$0.461600\pi$
$18$	0	0
$19$	−10.3521	−0.124997	−0.0624985	−	0.998045i	$-0.519907\pi$
$19$	−10.3521	−0.124997	−0.0624985	+	0.998045i	$0.519907\pi$
$20$	−79.4817	−0.888632
$21$	0	0
$22$	−37.4817	−0.363233
$23$	49.8704	0.452118	0.226059	−	0.974114i	$-0.427416\pi$
$23$	49.8704	0.452118	0.226059	+	0.974114i	$0.427416\pi$
$24$	0	0
$25$	269.834	2.15867
$26$	−91.7409	−0.691995
$27$	0	0
$28$	−23.4817	−0.158487
$29$	10.9070	0.0698408	0.0349204	−	0.999390i	$-0.488882\pi$
$29$	10.9070	0.0698408	0.0349204	+	0.999390i	$0.488882\pi$
$30$	0	0
$31$	151.611	0.878393	0.439197	−	0.898391i	$-0.355263\pi$
$31$	151.611	0.878393	0.439197	+	0.898391i	$0.355263\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−33.7409	−0.170191
$35$	116.648	0.563345
$36$	0	0
$37$	346.186	1.53818	0.769089	−	0.639141i	$-0.220710\pi$
$37$	346.186	1.53818	0.769089	+	0.639141i	$0.220710\pi$
$38$	20.7043	0.0883862
$39$	0	0
$40$	158.963	0.628358
$41$	264.741	1.00843	0.504214	−	0.863579i	$-0.331782\pi$
$41$	264.741	1.00843	0.504214	+	0.863579i	$0.331782\pi$
$42$	0	0
$43$	−411.890	−1.46076	−0.730380	−	0.683041i	$-0.760657\pi$
$43$	−411.890	−1.46076	−0.730380	+	0.683041i	$0.760657\pi$
$44$	74.9634	0.256845
$45$	0	0
$46$	−99.7409	−0.319695
$47$	472.056	1.46503	0.732516	−	0.680750i	$-0.238346\pi$
$47$	472.056	1.46503	0.732516	+	0.680750i	$0.238346\pi$
$48$	0	0
$49$	−308.538	−0.899528
$50$	−539.668	−1.52641
$51$	0	0
$52$	183.482	0.489314
$53$	290.186	0.752078	0.376039	−	0.926604i	$-0.377286\pi$
$53$	290.186	0.752078	0.376039	+	0.926604i	$0.377286\pi$
$54$	0	0
$55$	−372.389	−0.912962
$56$	46.9634	0.112067
$57$	0	0
$58$	−21.8140	−0.0493849
$59$	53.2591	0.117521	0.0587606	−	0.998272i	$-0.481285\pi$
$59$	53.2591	0.117521	0.0587606	+	0.998272i	$0.481285\pi$
$60$	0	0
$61$	−293.944	−0.616977	−0.308489	−	0.951228i	$-0.599823\pi$
$61$	−293.944	−0.616977	−0.308489	+	0.951228i	$0.599823\pi$
$62$	−303.223	−0.621118
$63$	0	0
$64$	64.0000	0.125000
$65$	−911.465	−1.73928
$66$	0	0
$67$	−398.555	−0.726735	−0.363367	−	0.931646i	$-0.618373\pi$
$67$	−398.555	−0.726735	−0.363367	+	0.931646i	$0.618373\pi$
$68$	67.4817	0.120344
$69$	0	0
$70$	−233.296	−0.398345
$71$	−647.854	−1.08290	−0.541451	−	0.840732i	$-0.682125\pi$
$71$	−647.854	−1.08290	−0.541451	+	0.840732i	$0.682125\pi$
$72$	0	0
$73$	−478.279	−0.766826	−0.383413	−	0.923577i	$-0.625251\pi$
$73$	−478.279	−0.766826	−0.383413	+	0.923577i	$0.625251\pi$
$74$	−692.372	−1.08766
$75$	0	0
$76$	−41.4085	−0.0624985
$77$	−110.017	−0.162826
$78$	0	0
$79$	374.316	0.533086	0.266543	−	0.963823i	$-0.414119\pi$
$79$	374.316	0.533086	0.266543	+	0.963823i	$0.414119\pi$
$80$	−317.927	−0.444316
$81$	0	0
$82$	−529.482	−0.713067
$83$	933.279	1.23422	0.617112	−	0.786875i	$-0.288302\pi$
$83$	933.279	1.23422	0.617112	+	0.786875i	$0.288302\pi$
$84$	0	0
$85$	−335.223	−0.427765
$86$	823.780	1.03291
$87$	0	0
$88$	−149.927	−0.181617
$89$	368.817	0.439264	0.219632	−	0.975583i	$-0.429514\pi$
$89$	368.817	0.439264	0.219632	+	0.975583i	$0.429514\pi$
$90$	0	0
$91$	−269.279	−0.310199
$92$	199.482	0.226059
$93$	0	0
$94$	−944.113	−1.03593
$95$	205.701	0.222153
$96$	0	0
$97$	274.149	0.286965	0.143483	−	0.989653i	$-0.454170\pi$
$97$	274.149	0.286965	0.143483	+	0.989653i	$0.454170\pi$
$98$	617.076	0.636062
$99$	0	0
$100$	1079.34	1.07934
$101$	−9.57168	−0.00942987	−0.00471494	−	0.999989i	$-0.501501\pi$
$101$	−9.57168	−0.00942987	−0.00471494	+	0.999989i	$0.501501\pi$
$102$	0	0
$103$	1971.50	1.88600	0.943001	−	0.332791i	$-0.107990\pi$
$103$	1971.50	1.88600	0.943001	+	0.332791i	$0.107990\pi$
$104$	−366.963	−0.345997
$105$	0	0
$106$	−580.372	−0.531799
$107$	1441.13	1.30205	0.651025	−	0.759057i	$-0.274339\pi$
$107$	1441.13	1.30205	0.651025	+	0.759057i	$0.274339\pi$
$108$	0	0
$109$	−90.3323	−0.0793786	−0.0396893	−	0.999212i	$-0.512637\pi$
$109$	−90.3323	−0.0793786	−0.0396893	+	0.999212i	$0.512637\pi$
$110$	744.777	0.645561
$111$	0	0
$112$	−93.9268	−0.0792433
$113$	−1650.73	−1.37422	−0.687112	−	0.726551i	$-0.741122\pi$
$113$	−1650.73	−1.37422	−0.687112	+	0.726551i	$0.741122\pi$
$114$	0	0
$115$	−990.947	−0.803533
$116$	43.6281	0.0349204
$117$	0	0
$118$	−106.518	−0.0831000
$119$	−99.0366	−0.0762913
$120$	0	0
$121$	−979.780	−0.736124
$122$	587.887	0.436269
$123$	0	0
$124$	606.445	0.439197
$125$	−2877.91	−2.05926
$126$	0	0
$127$	1997.45	1.39563	0.697814	−	0.716279i	$-0.254156\pi$
$127$	1997.45	1.39563	0.697814	+	0.716279i	$0.254156\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	1822.93	1.22986
$131$	−123.645	−0.0824649	−0.0412324	−	0.999150i	$-0.513128\pi$
$131$	−123.645	−0.0824649	−0.0412324	+	0.999150i	$0.513128\pi$
$132$	0	0
$133$	60.7714	0.0396207
$134$	797.110	0.513879
$135$	0	0
$136$	−134.963	−0.0850957
$137$	113.406	0.0707218	0.0353609	−	0.999375i	$-0.488742\pi$
$137$	113.406	0.0707218	0.0353609	+	0.999375i	$0.488742\pi$
$138$	0	0
$139$	−393.598	−0.240176	−0.120088	−	0.992763i	$-0.538318\pi$
$139$	−393.598	−0.240176	−0.120088	+	0.992763i	$0.538318\pi$
$140$	466.591	0.281673
$141$	0	0
$142$	1295.71	0.765728
$143$	859.651	0.502711
$144$	0	0
$145$	−216.727	−0.124126
$146$	956.558	0.542228
$147$	0	0
$148$	1384.74	0.769089
$149$	1507.21	0.828695	0.414348	−	0.910119i	$-0.364010\pi$
$149$	1507.21	0.828695	0.414348	+	0.910119i	$0.364010\pi$
$150$	0	0
$151$	3161.40	1.70378	0.851890	−	0.523720i	$-0.175456\pi$
$151$	3161.40	1.70378	0.851890	+	0.523720i	$0.175456\pi$
$152$	82.8170	0.0441931
$153$	0	0
$154$	220.034	0.115135
$155$	−3012.58	−1.56114
$156$	0	0
$157$	1375.61	0.699272	0.349636	−	0.936886i	$-0.386305\pi$
$157$	1375.61	0.699272	0.349636	+	0.936886i	$0.386305\pi$
$158$	−748.631	−0.376949
$159$	0	0
$160$	635.854	0.314179
$161$	−292.761	−0.143309
$162$	0	0
$163$	542.073	0.260481	0.130241	−	0.991482i	$-0.458425\pi$
$163$	542.073	0.260481	0.130241	+	0.991482i	$0.458425\pi$
$164$	1058.96	0.504214
$165$	0	0
$166$	−1866.56	−0.872729
$167$	3129.62	1.45016	0.725081	−	0.688664i	$-0.241802\pi$
$167$	3129.62	1.45016	0.725081	+	0.688664i	$0.241802\pi$
$168$	0	0
$169$	−92.9040	−0.0422868
$170$	670.445	0.302475
$171$	0	0
$172$	−1647.56	−0.730380
$173$	−2518.28	−1.10671	−0.553355	−	0.832945i	$-0.686653\pi$
$173$	−2518.28	−1.10671	−0.553355	+	0.832945i	$0.686653\pi$
$174$	0	0
$175$	−1584.04	−0.684241
$176$	299.854	0.128422
$177$	0	0
$178$	−737.634	−0.310607
$179$	−2331.26	−0.973443	−0.486721	−	0.873557i	$-0.661807\pi$
$179$	−2331.26	−0.973443	−0.486721	+	0.873557i	$0.661807\pi$
$180$	0	0
$181$	734.969	0.301822	0.150911	−	0.988547i	$-0.451779\pi$
$181$	734.969	0.301822	0.150911	+	0.988547i	$0.451779\pi$
$182$	538.558	0.219344
$183$	0	0
$184$	−398.963	−0.159848
$185$	−6878.86	−2.73375
$186$	0	0
$187$	316.166	0.123638
$188$	1888.23	0.732516
$189$	0	0
$190$	−411.402	−0.157086
$191$	3617.76	1.37053	0.685266	−	0.728293i	$-0.259686\pi$
$191$	3617.76	1.37053	0.685266	+	0.728293i	$0.259686\pi$
$192$	0	0
$193$	4341.12	1.61907	0.809536	−	0.587070i	$-0.199719\pi$
$193$	4341.12	1.61907	0.809536	+	0.587070i	$0.199719\pi$
$194$	−548.299	−0.202915
$195$	0	0
$196$	−1234.15	−0.449764
$197$	−3286.20	−1.18849	−0.594243	−	0.804285i	$-0.702548\pi$
$197$	−3286.20	−1.18849	−0.594243	+	0.804285i	$0.702548\pi$
$198$	0	0
$199$	332.265	0.118360	0.0591800	−	0.998247i	$-0.481151\pi$
$199$	332.265	0.118360	0.0591800	+	0.998247i	$0.481151\pi$
$200$	−2158.67	−0.763205
$201$	0	0
$202$	19.1434	0.00666793
$203$	−64.0288	−0.0221377
$204$	0	0
$205$	−5260.51	−1.79224
$206$	−3943.01	−1.33360
$207$	0	0
$208$	733.927	0.244657
$209$	−194.008	−0.0642096
$210$	0	0
$211$	−5742.47	−1.87359	−0.936797	−	0.349874i	$-0.886224\pi$
$211$	−5742.47	−1.87359	−0.936797	+	0.349874i	$0.886224\pi$
$212$	1160.74	0.376039
$213$	0	0
$214$	−2882.26	−0.920688
$215$	8184.43	2.59616
$216$	0	0
$217$	−890.023	−0.278427
$218$	180.665	0.0561291
$219$	0	0
$220$	−1489.55	−0.456481
$221$	773.854	0.235543
$222$	0	0
$223$	−2462.61	−0.739502	−0.369751	−	0.929131i	$-0.620557\pi$
$223$	−2462.61	−0.739502	−0.369751	+	0.929131i	$0.620557\pi$
$224$	187.854	0.0560335
$225$	0	0
$226$	3301.45	0.971723
$227$	2799.34	0.818496	0.409248	−	0.912423i	$-0.365791\pi$
$227$	2799.34	0.818496	0.409248	+	0.912423i	$0.365791\pi$
$228$	0	0
$229$	1412.85	0.407701	0.203850	−	0.979002i	$-0.434654\pi$
$229$	1412.85	0.407701	0.203850	+	0.979002i	$0.434654\pi$
$230$	1981.89	0.568183
$231$	0	0
$232$	−87.2561	−0.0246924
$233$	2033.81	0.571843	0.285922	−	0.958253i	$-0.407700\pi$
$233$	2033.81	0.571843	0.285922	+	0.958253i	$0.407700\pi$
$234$	0	0
$235$	−9379.96	−2.60375
$236$	213.037	0.0587606
$237$	0	0
$238$	198.073	0.0539461
$239$	−521.611	−0.141173	−0.0705863	−	0.997506i	$-0.522487\pi$
$239$	−521.611	−0.141173	−0.0705863	+	0.997506i	$0.522487\pi$
$240$	0	0
$241$	5915.65	1.58116	0.790582	−	0.612356i	$-0.209778\pi$
$241$	5915.65	1.58116	0.790582	+	0.612356i	$0.209778\pi$
$242$	1959.56	0.520518
$243$	0	0
$244$	−1175.77	−0.308489
$245$	6130.78	1.59870
$246$	0	0
$247$	−474.857	−0.122326
$248$	−1212.89	−0.310559
$249$	0	0
$250$	5755.82	1.45612
$251$	710.127	0.178577	0.0892884	−	0.996006i	$-0.471541\pi$
$251$	710.127	0.178577	0.0892884	+	0.996006i	$0.471541\pi$
$252$	0	0
$253$	934.614	0.232248
$254$	−3994.90	−0.986859
$255$	0	0
$256$	256.000	0.0625000
$257$	6301.01	1.52936	0.764681	−	0.644409i	$-0.222897\pi$
$257$	6301.01	1.52936	0.764681	+	0.644409i	$0.222897\pi$
$258$	0	0
$259$	−2032.26	−0.487562
$260$	−3645.86	−0.869641
$261$	0	0
$262$	247.290	0.0583115
$263$	−4921.82	−1.15396	−0.576981	−	0.816757i	$-0.695769\pi$
$263$	−4921.82	−1.15396	−0.576981	+	0.816757i	$0.695769\pi$
$264$	0	0
$265$	−5766.12	−1.33664
$266$	−121.543	−0.0280161
$267$	0	0
$268$	−1594.22	−0.363367
$269$	−5454.50	−1.23631	−0.618154	−	0.786057i	$-0.712119\pi$
$269$	−5454.50	−1.23631	−0.618154	+	0.786057i	$0.712119\pi$
$270$	0	0
$271$	2797.10	0.626981	0.313491	−	0.949591i	$-0.398502\pi$
$271$	2797.10	0.626981	0.313491	+	0.949591i	$0.398502\pi$
$272$	269.927	0.0601718
$273$	0	0
$274$	−226.811	−0.0500079
$275$	5056.92	1.10889
$276$	0	0
$277$	−2145.17	−0.465310	−0.232655	−	0.972559i	$-0.574741\pi$
$277$	−2145.17	−0.465310	−0.232655	+	0.972559i	$0.574741\pi$
$278$	787.195	0.169830
$279$	0	0
$280$	−933.183	−0.199173
$281$	−5509.77	−1.16970	−0.584849	−	0.811142i	$-0.698846\pi$
$281$	−5509.77	−1.16970	−0.584849	+	0.811142i	$0.698846\pi$
$282$	0	0
$283$	4007.84	0.841842	0.420921	−	0.907097i	$-0.361707\pi$
$283$	4007.84	0.841842	0.420921	+	0.907097i	$0.361707\pi$
$284$	−2591.41	−0.541451
$285$	0	0
$286$	−1719.30	−0.355470
$287$	−1554.14	−0.319645
$288$	0	0
$289$	−4628.39	−0.942070
$290$	433.454	0.0877700
$291$	0	0
$292$	−1913.12	−0.383413
$293$	−5809.24	−1.15829	−0.579146	−	0.815224i	$-0.696614\pi$
$293$	−5809.24	−1.15829	−0.579146	+	0.815224i	$0.696614\pi$
$294$	0	0
$295$	−1058.28	−0.208866
$296$	−2769.49	−0.543828
$297$	0	0
$298$	−3014.42	−0.585976
$299$	2287.58	0.442455
$300$	0	0
$301$	2417.97	0.463022
$302$	−6322.80	−1.20475
$303$	0	0
$304$	−165.634	−0.0312492
$305$	5840.78	1.09653
$306$	0	0
$307$	8688.30	1.61520	0.807602	−	0.589728i	$-0.200765\pi$
$307$	8688.30	1.61520	0.807602	+	0.589728i	$0.200765\pi$
$308$	−440.067	−0.0814128
$309$	0	0
$310$	6025.16	1.10389
$311$	1432.01	0.261099	0.130550	−	0.991442i	$-0.458326\pi$
$311$	1432.01	0.261099	0.130550	+	0.991442i	$0.458326\pi$
$312$	0	0
$313$	6614.52	1.19449	0.597244	−	0.802059i	$-0.296262\pi$
$313$	6614.52	1.19449	0.597244	+	0.802059i	$0.296262\pi$
$314$	−2751.22	−0.494460
$315$	0	0
$316$	1497.26	0.266543
$317$	1424.09	0.252318	0.126159	−	0.992010i	$-0.459735\pi$
$317$	1424.09	0.252318	0.126159	+	0.992010i	$0.459735\pi$
$318$	0	0
$319$	204.407	0.0358764
$320$	−1271.71	−0.222158
$321$	0	0
$322$	585.521	0.101335
$323$	−174.645	−0.0300851
$324$	0	0
$325$	12377.4	2.11254
$326$	−1084.15	−0.184188
$327$	0	0
$328$	−2117.93	−0.356533
$329$	−2771.17	−0.464376
$330$	0	0
$331$	3537.65	0.587453	0.293726	−	0.955890i	$-0.405105\pi$
$331$	3537.65	0.587453	0.293726	+	0.955890i	$0.405105\pi$
$332$	3733.12	0.617112
$333$	0	0
$334$	−6259.23	−1.02542
$335$	7919.46	1.29160
$336$	0	0
$337$	−1760.28	−0.284536	−0.142268	−	0.989828i	$-0.545439\pi$
$337$	−1760.28	−0.284536	−0.142268	+	0.989828i	$0.545439\pi$
$338$	185.808	0.0299013
$339$	0	0
$340$	−1340.89	−0.213882
$341$	2841.32	0.451221
$342$	0	0
$343$	3824.81	0.602099
$344$	3295.12	0.516457
$345$	0	0
$346$	5036.55	0.782563
$347$	−1605.44	−0.248370	−0.124185	−	0.992259i	$-0.539632\pi$
$347$	−1605.44	−0.248370	−0.124185	+	0.992259i	$0.539632\pi$
$348$	0	0
$349$	−6641.89	−1.01872	−0.509358	−	0.860555i	$-0.670117\pi$
$349$	−6641.89	−1.01872	−0.509358	+	0.860555i	$0.670117\pi$
$350$	3168.08	0.483831
$351$	0	0
$352$	−599.707	−0.0908083
$353$	−3056.27	−0.460818	−0.230409	−	0.973094i	$-0.574006\pi$
$353$	−3056.27	−0.460818	−0.230409	+	0.973094i	$0.574006\pi$
$354$	0	0
$355$	12873.1	1.92460
$356$	1475.27	0.219632
$357$	0	0
$358$	4662.51	0.688328
$359$	−2489.46	−0.365985	−0.182993	−	0.983114i	$-0.558578\pi$
$359$	−2489.46	−0.365985	−0.182993	+	0.983114i	$0.558578\pi$
$360$	0	0
$361$	−6751.83	−0.984376
$362$	−1469.94	−0.213421
$363$	0	0
$364$	−1077.12	−0.155099
$365$	9503.61	1.36285
$366$	0	0
$367$	2355.05	0.334966	0.167483	−	0.985875i	$-0.446436\pi$
$367$	2355.05	0.334966	0.167483	+	0.985875i	$0.446436\pi$
$368$	797.927	0.113029
$369$	0	0
$370$	13757.7	1.93305
$371$	−1703.52	−0.238388
$372$	0	0
$373$	−2757.03	−0.382717	−0.191358	−	0.981520i	$-0.561289\pi$
$373$	−2757.03	−0.382717	−0.191358	+	0.981520i	$0.561289\pi$
$374$	−632.332	−0.0874255
$375$	0	0
$376$	−3776.45	−0.517967
$377$	500.310	0.0683481
$378$	0	0
$379$	246.459	0.0334030	0.0167015	−	0.999861i	$-0.494683\pi$
$379$	246.459	0.0334030	0.0167015	+	0.999861i	$0.494683\pi$
$380$	822.805	0.111076
$381$	0	0
$382$	−7235.52	−0.969113
$383$	650.575	0.0867959	0.0433979	−	0.999058i	$-0.486182\pi$
$383$	650.575	0.0867959	0.0433979	+	0.999058i	$0.486182\pi$
$384$	0	0
$385$	2186.08	0.289384
$386$	−8682.25	−1.14486
$387$	0	0
$388$	1096.60	0.143483
$389$	−10246.2	−1.33548	−0.667739	−	0.744395i	$-0.732738\pi$
$389$	−10246.2	−1.33548	−0.667739	+	0.744395i	$0.732738\pi$
$390$	0	0
$391$	841.335	0.108819
$392$	2468.30	0.318031
$393$	0	0
$394$	6572.40	0.840387
$395$	−7437.81	−0.947435
$396$	0	0
$397$	−9453.68	−1.19513	−0.597565	−	0.801820i	$-0.703865\pi$
$397$	−9453.68	−1.19513	−0.597565	+	0.801820i	$0.703865\pi$
$398$	−664.530	−0.0836932
$399$	0	0
$400$	4317.34	0.539668
$401$	270.723	0.0337138	0.0168569	−	0.999858i	$-0.494634\pi$
$401$	270.723	0.0337138	0.0168569	+	0.999858i	$0.494634\pi$
$402$	0	0
$403$	6954.47	0.859620
$404$	−38.2867	−0.00471494
$405$	0	0
$406$	128.058	0.0156537
$407$	6487.82	0.790146
$408$	0	0
$409$	−11586.2	−1.40073	−0.700366	−	0.713784i	$-0.746980\pi$
$409$	−11586.2	−1.40073	−0.700366	+	0.713784i	$0.746980\pi$
$410$	10521.0	1.26731
$411$	0	0
$412$	7886.02	0.943001
$413$	−312.654	−0.0372511
$414$	0	0
$415$	−18544.7	−2.19354
$416$	−1467.85	−0.172999
$417$	0	0
$418$	388.016	0.0454030
$419$	−16133.7	−1.88111	−0.940554	−	0.339644i	$-0.889693\pi$
$419$	−16133.7	−1.88111	−0.940554	+	0.339644i	$0.889693\pi$
$420$	0	0
$421$	4991.93	0.577890	0.288945	−	0.957346i	$-0.406696\pi$
$421$	4991.93	0.577890	0.288945	+	0.957346i	$0.406696\pi$
$422$	11484.9	1.32483
$423$	0	0
$424$	−2321.49	−0.265900
$425$	4552.21	0.519564
$426$	0	0
$427$	1725.57	0.195565
$428$	5764.52	0.651025
$429$	0	0
$430$	−16368.9	−1.83576
$431$	8184.74	0.914722	0.457361	−	0.889281i	$-0.348795\pi$
$431$	8184.74	0.914722	0.457361	+	0.889281i	$0.348795\pi$
$432$	0	0
$433$	8663.17	0.961490	0.480745	−	0.876860i	$-0.340366\pi$
$433$	8663.17	0.961490	0.480745	+	0.876860i	$0.340366\pi$
$434$	1780.05	0.196878
$435$	0	0
$436$	−361.329	−0.0396893
$437$	−516.265	−0.0565133
$438$	0	0
$439$	−15864.3	−1.72474	−0.862371	−	0.506277i	$-0.831021\pi$
$439$	−15864.3	−1.72474	−0.862371	+	0.506277i	$0.831021\pi$
$440$	2979.11	0.322781
$441$	0	0
$442$	−1547.71	−0.166554
$443$	−399.641	−0.0428612	−0.0214306	−	0.999770i	$-0.506822\pi$
$443$	−399.641	−0.0428612	−0.0214306	+	0.999770i	$0.506822\pi$
$444$	0	0
$445$	−7328.55	−0.780689
$446$	4925.23	0.522907
$447$	0	0
$448$	−375.707	−0.0396217
$449$	5924.51	0.622706	0.311353	−	0.950294i	$-0.399218\pi$
$449$	5924.51	0.622706	0.311353	+	0.950294i	$0.399218\pi$
$450$	0	0
$451$	4961.47	0.518019
$452$	−6602.91	−0.687112
$453$	0	0
$454$	−5598.68	−0.578764
$455$	5350.69	0.551306
$456$	0	0
$457$	−7891.14	−0.807728	−0.403864	−	0.914819i	$-0.632333\pi$
$457$	−7891.14	−0.807728	−0.403864	+	0.914819i	$0.632333\pi$
$458$	−2825.69	−0.288288
$459$	0	0
$460$	−3963.79	−0.401766
$461$	−3262.18	−0.329577	−0.164789	−	0.986329i	$-0.552694\pi$
$461$	−3262.18	−0.329577	−0.164789	+	0.986329i	$0.552694\pi$
$462$	0	0
$463$	3690.40	0.370427	0.185213	−	0.982698i	$-0.440702\pi$
$463$	3690.40	0.370427	0.185213	+	0.982698i	$0.440702\pi$
$464$	174.512	0.0174602
$465$	0	0
$466$	−4067.62	−0.404354
$467$	−10193.2	−1.01003	−0.505017	−	0.863110i	$-0.668514\pi$
$467$	−10193.2	−1.01003	−0.505017	+	0.863110i	$0.668514\pi$
$468$	0	0
$469$	2339.69	0.230355
$470$	18759.9	1.84113
$471$	0	0
$472$	−426.073	−0.0415500
$473$	−7719.17	−0.750376
$474$	0	0
$475$	−2793.36	−0.269827
$476$	−396.146	−0.0381457
$477$	0	0
$478$	1043.22	0.0998240
$479$	840.068	0.0801330	0.0400665	−	0.999197i	$-0.487243\pi$
$479$	840.068	0.0801330	0.0400665	+	0.999197i	$0.487243\pi$
$480$	0	0
$481$	15879.7	1.50531
$482$	−11831.3	−1.11805
$483$	0	0
$484$	−3919.12	−0.368062
$485$	−5447.46	−0.510014
$486$	0	0
$487$	−3367.28	−0.313319	−0.156659	−	0.987653i	$-0.550072\pi$
$487$	−3367.28	−0.313319	−0.156659	+	0.987653i	$0.550072\pi$
$488$	2351.55	0.218134
$489$	0	0
$490$	−12261.6	−1.13045
$491$	−18869.1	−1.73432	−0.867160	−	0.498030i	$-0.834057\pi$
$491$	−18869.1	−1.73432	−0.867160	+	0.498030i	$0.834057\pi$
$492$	0	0
$493$	184.006	0.0168098
$494$	949.713	0.0864972
$495$	0	0
$496$	2425.78	0.219598
$497$	3803.18	0.343251
$498$	0	0
$499$	−14566.0	−1.30674	−0.653371	−	0.757038i	$-0.726646\pi$
$499$	−14566.0	−1.30674	−0.653371	+	0.757038i	$0.726646\pi$
$500$	−11511.6	−1.02963
$501$	0	0
$502$	−1420.25	−0.126273
$503$	−17361.6	−1.53900	−0.769499	−	0.638648i	$-0.779494\pi$
$503$	−17361.6	−1.53900	−0.769499	+	0.638648i	$0.779494\pi$
$504$	0	0
$505$	190.193	0.0167594
$506$	−1869.23	−0.164224
$507$	0	0
$508$	7989.79	0.697814
$509$	13933.3	1.21332	0.606661	−	0.794961i	$-0.292509\pi$
$509$	13933.3	1.21332	0.606661	+	0.794961i	$0.292509\pi$
$510$	0	0
$511$	2807.70	0.243063
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−12602.0	−1.08142
$515$	−39174.6	−3.35192
$516$	0	0
$517$	8846.74	0.752571
$518$	4064.52	0.344758
$519$	0	0
$520$	7291.72	0.614929
$521$	5024.22	0.422486	0.211243	−	0.977434i	$-0.432249\pi$
$521$	5024.22	0.422486	0.211243	+	0.977434i	$0.432249\pi$
$522$	0	0
$523$	16008.7	1.33845	0.669226	−	0.743059i	$-0.266626\pi$
$523$	16008.7	1.33845	0.669226	+	0.743059i	$0.266626\pi$
$524$	−494.579	−0.0412324
$525$	0	0
$526$	9843.63	0.815975
$527$	2557.75	0.211418
$528$	0	0
$529$	−9679.94	−0.795590
$530$	11532.2	0.945148
$531$	0	0
$532$	243.086	0.0198103
$533$	12143.8	0.986877
$534$	0	0
$535$	−28635.9	−2.31409
$536$	3188.44	0.256940
$537$	0	0
$538$	10909.0	0.874202
$539$	−5782.27	−0.462078
$540$	0	0
$541$	−10094.1	−0.802179	−0.401089	−	0.916039i	$-0.631368\pi$
$541$	−10094.1	−0.802179	−0.401089	+	0.916039i	$0.631368\pi$
$542$	−5594.21	−0.443343
$543$	0	0
$544$	−539.854	−0.0425479
$545$	1794.94	0.141077
$546$	0	0
$547$	−6060.54	−0.473730	−0.236865	−	0.971543i	$-0.576120\pi$
$547$	−6060.54	−0.473730	−0.236865	+	0.971543i	$0.576120\pi$
$548$	453.622	0.0353609
$549$	0	0
$550$	−10113.8	−0.784100
$551$	−112.911	−0.00872988
$552$	0	0
$553$	−2197.39	−0.168974
$554$	4290.34	0.329024
$555$	0	0
$556$	−1574.39	−0.120088
$557$	13688.4	1.04128	0.520642	−	0.853775i	$-0.325693\pi$
$557$	13688.4	1.04128	0.520642	+	0.853775i	$0.325693\pi$
$558$	0	0
$559$	−18893.6	−1.42954
$560$	1866.37	0.140836
$561$	0	0
$562$	11019.5	0.827102
$563$	22346.6	1.67282	0.836411	−	0.548103i	$-0.184650\pi$
$563$	22346.6	1.67282	0.836411	+	0.548103i	$0.184650\pi$
$564$	0	0
$565$	32800.7	2.44236
$566$	−8015.67	−0.595272
$567$	0	0
$568$	5182.83	0.382864
$569$	8117.94	0.598105	0.299052	−	0.954237i	$-0.403329\pi$
$569$	8117.94	0.598105	0.299052	+	0.954237i	$0.403329\pi$
$570$	0	0
$571$	6982.97	0.511783	0.255892	−	0.966705i	$-0.417631\pi$
$571$	6982.97	0.511783	0.255892	+	0.966705i	$0.417631\pi$
$572$	3438.60	0.251355
$573$	0	0
$574$	3108.28	0.226023
$575$	13456.7	0.975973
$576$	0	0
$577$	13972.4	1.00811	0.504055	−	0.863671i	$-0.331841\pi$
$577$	13972.4	1.00811	0.504055	+	0.863671i	$0.331841\pi$
$578$	9256.78	0.666144
$579$	0	0
$580$	−866.908	−0.0620628
$581$	−5478.75	−0.391216
$582$	0	0
$583$	5438.33	0.386334
$584$	3826.23	0.271114
$585$	0	0
$586$	11618.5	0.819036
$587$	−1509.51	−0.106140	−0.0530701	−	0.998591i	$-0.516901\pi$
$587$	−1509.51	−0.106140	−0.0530701	+	0.998591i	$0.516901\pi$
$588$	0	0
$589$	−1569.50	−0.109796
$590$	2116.56	0.147691
$591$	0	0
$592$	5538.98	0.384545
$593$	−21495.3	−1.48854	−0.744270	−	0.667879i	$-0.767202\pi$
$593$	−21495.3	−1.48854	−0.744270	+	0.667879i	$0.767202\pi$
$594$	0	0
$595$	1967.90	0.135590
$596$	6028.85	0.414348
$597$	0	0
$598$	−4575.16	−0.312863
$599$	10104.9	0.689272	0.344636	−	0.938736i	$-0.388002\pi$
$599$	10104.9	0.689272	0.344636	+	0.938736i	$0.388002\pi$
$600$	0	0
$601$	−11008.2	−0.747143	−0.373572	−	0.927601i	$-0.621867\pi$
$601$	−11008.2	−0.747143	−0.373572	+	0.927601i	$0.621867\pi$
$602$	−4835.94	−0.327406
$603$	0	0
$604$	12645.6	0.851890
$605$	19468.7	1.30829
$606$	0	0
$607$	14680.2	0.981629	0.490815	−	0.871264i	$-0.336699\pi$
$607$	14680.2	0.981629	0.490815	+	0.871264i	$0.336699\pi$
$608$	331.268	0.0220965
$609$	0	0
$610$	−11681.6	−0.775365
$611$	21653.4	1.43372
$612$	0	0
$613$	235.863	0.0155407	0.00777033	−	0.999970i	$-0.497527\pi$
$613$	235.863	0.0155407	0.00777033	+	0.999970i	$0.497527\pi$
$614$	−17376.6	−1.14212
$615$	0	0
$616$	880.134	0.0575676
$617$	18313.1	1.19490	0.597452	−	0.801904i	$-0.296180\pi$
$617$	18313.1	1.19490	0.597452	+	0.801904i	$0.296180\pi$
$618$	0	0
$619$	6849.46	0.444755	0.222377	−	0.974961i	$-0.428618\pi$
$619$	6849.46	0.444755	0.222377	+	0.974961i	$0.428618\pi$
$620$	−12050.3	−0.780569
$621$	0	0
$622$	−2864.02	−0.184625
$623$	−2165.11	−0.139235
$624$	0	0
$625$	23456.1	1.50119
$626$	−13229.0	−0.844631
$627$	0	0
$628$	5502.45	0.349636
$629$	5840.30	0.370220
$630$	0	0
$631$	22464.3	1.41726	0.708629	−	0.705581i	$-0.249314\pi$
$631$	22464.3	1.41726	0.708629	+	0.705581i	$0.249314\pi$
$632$	−2994.52	−0.188474
$633$	0	0
$634$	−2848.18	−0.178416
$635$	−39690.1	−2.48040
$636$	0	0
$637$	−14152.8	−0.880303
$638$	−408.814	−0.0253685
$639$	0	0
$640$	2543.41	0.157090
$641$	18888.8	1.16390	0.581952	−	0.813223i	$-0.302289\pi$
$641$	18888.8	1.16390	0.581952	+	0.813223i	$0.302289\pi$
$642$	0	0
$643$	6793.95	0.416683	0.208342	−	0.978056i	$-0.433193\pi$
$643$	6793.95	0.416683	0.208342	+	0.978056i	$0.433193\pi$
$644$	−1171.04	−0.0716546
$645$	0	0
$646$	349.290	0.0212734
$647$	5277.92	0.320706	0.160353	−	0.987060i	$-0.448737\pi$
$647$	5277.92	0.320706	0.160353	+	0.987060i	$0.448737\pi$
$648$	0	0
$649$	998.122	0.0603694
$650$	−24754.8	−1.49379
$651$	0	0
$652$	2168.29	0.130241
$653$	−16564.2	−0.992659	−0.496330	−	0.868134i	$-0.665319\pi$
$653$	−16564.2	−0.992659	−0.496330	+	0.868134i	$0.665319\pi$
$654$	0	0
$655$	2456.88	0.146562
$656$	4235.85	0.252107
$657$	0	0
$658$	5542.34	0.328363
$659$	−3301.02	−0.195128	−0.0975641	−	0.995229i	$-0.531105\pi$
$659$	−3301.02	−0.195128	−0.0975641	+	0.995229i	$0.531105\pi$
$660$	0	0
$661$	12990.3	0.764395	0.382198	−	0.924081i	$-0.375167\pi$
$661$	12990.3	0.764395	0.382198	+	0.924081i	$0.375167\pi$
$662$	−7075.30	−0.415392
$663$	0	0
$664$	−7466.23	−0.436364
$665$	−1207.55	−0.0704164
$666$	0	0
$667$	543.938	0.0315762
$668$	12518.5	0.725081
$669$	0	0
$670$	−15838.9	−0.913299
$671$	−5508.75	−0.316935
$672$	0	0
$673$	18514.3	1.06044	0.530219	−	0.847861i	$-0.322110\pi$
$673$	18514.3	1.06044	0.530219	+	0.847861i	$0.322110\pi$
$674$	3520.56	0.201197
$675$	0	0
$676$	−371.616	−0.0211434
$677$	−6099.99	−0.346295	−0.173148	−	0.984896i	$-0.555394\pi$
$677$	−6099.99	−0.346295	−0.173148	+	0.984896i	$0.555394\pi$
$678$	0	0
$679$	−1609.37	−0.0909604
$680$	2681.78	0.151238
$681$	0	0
$682$	−5682.65	−0.319061
$683$	−22630.0	−1.26781	−0.633905	−	0.773411i	$-0.718549\pi$
$683$	−22630.0	−1.26781	−0.633905	+	0.773411i	$0.718549\pi$
$684$	0	0
$685$	−2253.42	−0.125691
$686$	−7649.61	−0.425749
$687$	0	0
$688$	−6590.24	−0.365190
$689$	13311.0	0.736004
$690$	0	0
$691$	22986.8	1.26550	0.632750	−	0.774356i	$-0.281926\pi$
$691$	22986.8	1.26550	0.632750	+	0.774356i	$0.281926\pi$
$692$	−10073.1	−0.553355
$693$	0	0
$694$	3210.88	0.175624
$695$	7820.95	0.426857
$696$	0	0
$697$	4466.29	0.242716
$698$	13283.8	0.720342
$699$	0	0
$700$	−6336.16	−0.342120
$701$	27015.5	1.45558	0.727790	−	0.685800i	$-0.240548\pi$
$701$	27015.5	1.45558	0.727790	+	0.685800i	$0.240548\pi$
$702$	0	0
$703$	−3583.76	−0.192268
$704$	1199.41	0.0642111
$705$	0	0
$706$	6112.54	0.325848
$707$	56.1898	0.00298902
$708$	0	0
$709$	−19177.3	−1.01582	−0.507912	−	0.861409i	$-0.669583\pi$
$709$	−19177.3	−1.01582	−0.507912	+	0.861409i	$0.669583\pi$
$710$	−25746.3	−1.36090
$711$	0	0
$712$	−2950.54	−0.155303
$713$	7560.92	0.397137
$714$	0	0
$715$	−17081.6	−0.893450
$716$	−9325.02	−0.486721
$717$	0	0
$718$	4978.92	0.258791
$719$	25931.8	1.34505	0.672526	−	0.740073i	$-0.265209\pi$
$719$	25931.8	1.34505	0.672526	+	0.740073i	$0.265209\pi$
$720$	0	0
$721$	−11573.6	−0.597812
$722$	13503.7	0.696059
$723$	0	0
$724$	2939.88	0.150911
$725$	2943.08	0.150763
$726$	0	0
$727$	5830.50	0.297443	0.148722	−	0.988879i	$-0.452484\pi$
$727$	5830.50	0.297443	0.148722	+	0.988879i	$0.452484\pi$
$728$	2154.23	0.109672
$729$	0	0
$730$	−19007.2	−0.963683
$731$	−6948.76	−0.351586
$732$	0	0
$733$	−23154.6	−1.16676	−0.583379	−	0.812200i	$-0.698270\pi$
$733$	−23154.6	−1.16676	−0.583379	+	0.812200i	$0.698270\pi$
$734$	−4710.09	−0.236857
$735$	0	0
$736$	−1595.85	−0.0799238
$737$	−7469.26	−0.373316
$738$	0	0
$739$	−27085.8	−1.34827	−0.674133	−	0.738610i	$-0.735483\pi$
$739$	−27085.8	−1.34827	−0.674133	+	0.738610i	$0.735483\pi$
$740$	−27515.5	−1.36688
$741$	0	0
$742$	3407.03	0.168566
$743$	32187.6	1.58930	0.794650	−	0.607068i	$-0.207655\pi$
$743$	32187.6	1.58930	0.794650	+	0.607068i	$0.207655\pi$
$744$	0	0
$745$	−29948.9	−1.47281
$746$	5514.05	0.270622
$747$	0	0
$748$	1264.66	0.0618191
$749$	−8460.04	−0.412715
$750$	0	0
$751$	−2732.93	−0.132791	−0.0663954	−	0.997793i	$-0.521150\pi$
$751$	−2732.93	−0.132791	−0.0663954	+	0.997793i	$0.521150\pi$
$752$	7552.90	0.366258
$753$	0	0
$754$	−1000.62	−0.0483294
$755$	−62818.3	−3.02807
$756$	0	0
$757$	−6315.62	−0.303230	−0.151615	−	0.988440i	$-0.548447\pi$
$757$	−6315.62	−0.303230	−0.151615	+	0.988440i	$0.548447\pi$
$758$	−492.918	−0.0236195
$759$	0	0
$760$	−1645.61	−0.0785428
$761$	31481.3	1.49960	0.749801	−	0.661663i	$-0.230149\pi$
$761$	31481.3	1.49960	0.749801	+	0.661663i	$0.230149\pi$
$762$	0	0
$763$	530.289	0.0251609
$764$	14471.0	0.685266
$765$	0	0
$766$	−1301.15	−0.0613739
$767$	2443.02	0.115010
$768$	0	0
$769$	−27296.6	−1.28003	−0.640014	−	0.768363i	$-0.721071\pi$
$769$	−27296.6	−1.28003	−0.640014	+	0.768363i	$0.721071\pi$
$770$	−4372.16	−0.204626
$771$	0	0
$772$	17364.5	0.809536
$773$	24025.2	1.11789	0.558943	−	0.829206i	$-0.311207\pi$
$773$	24025.2	1.11789	0.558943	+	0.829206i	$0.311207\pi$
$774$	0	0
$775$	40909.9	1.89616
$776$	−2193.20	−0.101458
$777$	0	0
$778$	20492.3	0.944326
$779$	−2740.63	−0.126050
$780$	0	0
$781$	−12141.3	−0.556275
$782$	−1682.67	−0.0769465
$783$	0	0
$784$	−4936.61	−0.224882
$785$	−27334.0	−1.24279
$786$	0	0
$787$	−10558.9	−0.478251	−0.239125	−	0.970989i	$-0.576861\pi$
$787$	−10558.9	−0.478251	−0.239125	+	0.970989i	$0.576861\pi$
$788$	−13144.8	−0.594243
$789$	0	0
$790$	14875.6	0.669938
$791$	9690.47	0.435592
$792$	0	0
$793$	−13483.3	−0.603792
$794$	18907.4	0.845084
$795$	0	0
$796$	1329.06	0.0591800
$797$	−26889.1	−1.19506	−0.597529	−	0.801847i	$-0.703851\pi$
$797$	−26889.1	−1.19506	−0.597529	+	0.801847i	$0.703851\pi$
$798$	0	0
$799$	7963.79	0.352614
$800$	−8634.68	−0.381603
$801$	0	0
$802$	−541.446	−0.0238393
$803$	−8963.36	−0.393910
$804$	0	0
$805$	5817.28	0.254698
$806$	−13908.9	−0.607843
$807$	0	0
$808$	76.5734	0.00333396
$809$	−18643.8	−0.810236	−0.405118	−	0.914264i	$-0.632770\pi$
$809$	−18643.8	−0.810236	−0.405118	+	0.914264i	$0.632770\pi$
$810$	0	0
$811$	−23870.8	−1.03356	−0.516781	−	0.856118i	$-0.672870\pi$
$811$	−23870.8	−1.03356	−0.516781	+	0.856118i	$0.672870\pi$
$812$	−256.115	−0.0110688
$813$	0	0
$814$	−12975.6	−0.558717
$815$	−10771.2	−0.462944
$816$	0	0
$817$	4263.94	0.182590
$818$	23172.3	0.990467
$819$	0	0
$820$	−21042.1	−0.896122
$821$	14533.0	0.617791	0.308895	−	0.951096i	$-0.400041\pi$
$821$	14533.0	0.617791	0.308895	+	0.951096i	$0.400041\pi$
$822$	0	0
$823$	8868.42	0.375618	0.187809	−	0.982206i	$-0.439861\pi$
$823$	8868.42	0.375618	0.187809	+	0.982206i	$0.439861\pi$
$824$	−15772.0	−0.666802
$825$	0	0
$826$	625.308	0.0263405
$827$	25059.3	1.05369	0.526843	−	0.849963i	$-0.323376\pi$
$827$	25059.3	1.05369	0.526843	+	0.849963i	$0.323376\pi$
$828$	0	0
$829$	−22556.3	−0.945009	−0.472505	−	0.881328i	$-0.656650\pi$
$829$	−22556.3	−0.945009	−0.472505	+	0.881328i	$0.656650\pi$
$830$	37089.3	1.55107
$831$	0	0
$832$	2935.71	0.122329
$833$	−5205.17	−0.216505
$834$	0	0
$835$	−62186.8	−2.57732
$836$	−776.031	−0.0321048
$837$	0	0
$838$	32267.5	1.33014
$839$	21209.6	0.872748	0.436374	−	0.899765i	$-0.356262\pi$
$839$	21209.6	0.872748	0.436374	+	0.899765i	$0.356262\pi$
$840$	0	0
$841$	−24270.0	−0.995122
$842$	−9983.86	−0.408630
$843$	0	0
$844$	−22969.9	−0.936797
$845$	1846.04	0.0751548
$846$	0	0
$847$	5751.73	0.233331
$848$	4642.98	0.188019
$849$	0	0
$850$	−9104.42	−0.367387
$851$	17264.4	0.695438
$852$	0	0
$853$	28116.2	1.12858	0.564292	−	0.825576i	$-0.309149\pi$
$853$	28116.2	1.12858	0.564292	+	0.825576i	$0.309149\pi$
$854$	−3451.15	−0.138286
$855$	0	0
$856$	−11529.0	−0.460344
$857$	−9477.64	−0.377771	−0.188886	−	0.981999i	$-0.560488\pi$
$857$	−9477.64	−0.377771	−0.188886	+	0.981999i	$0.560488\pi$
$858$	0	0
$859$	29545.8	1.17356	0.586780	−	0.809746i	$-0.300395\pi$
$859$	29545.8	1.17356	0.586780	+	0.809746i	$0.300395\pi$
$860$	32737.7	1.29808
$861$	0	0
$862$	−16369.5	−0.646806
$863$	9932.53	0.391781	0.195891	−	0.980626i	$-0.437240\pi$
$863$	9932.53	0.391781	0.195891	+	0.980626i	$0.437240\pi$
$864$	0	0
$865$	50039.2	1.96692
$866$	−17326.3	−0.679876
$867$	0	0
$868$	−3560.09	−0.139214
$869$	7014.99	0.273840
$870$	0	0
$871$	−18281.9	−0.711203
$872$	722.659	0.0280646
$873$	0	0
$874$	1032.53	0.0399609
$875$	16894.6	0.652732
$876$	0	0
$877$	−27025.5	−1.04058	−0.520288	−	0.853991i	$-0.674175\pi$
$877$	−27025.5	−1.04058	−0.520288	+	0.853991i	$0.674175\pi$
$878$	31728.6	1.21958
$879$	0	0
$880$	−5958.22	−0.228240
$881$	−12798.1	−0.489422	−0.244711	−	0.969596i	$-0.578693\pi$
$881$	−12798.1	−0.489422	−0.244711	+	0.969596i	$0.578693\pi$
$882$	0	0
$883$	27016.7	1.02965	0.514826	−	0.857294i	$-0.327856\pi$
$883$	27016.7	1.02965	0.514826	+	0.857294i	$0.327856\pi$
$884$	3095.41	0.117772
$885$	0	0
$886$	799.281	0.0303074
$887$	−20150.1	−0.762768	−0.381384	−	0.924417i	$-0.624552\pi$
$887$	−20150.1	−0.762768	−0.381384	+	0.924417i	$0.624552\pi$
$888$	0	0
$889$	−11725.9	−0.442377
$890$	14657.1	0.552031
$891$	0	0
$892$	−9850.46	−0.369751
$893$	−4886.79	−0.183124
$894$	0	0
$895$	46323.1	1.73007
$896$	751.415	0.0280167
$897$	0	0
$898$	−11849.0	−0.440320
$899$	1653.63	0.0613477
$900$	0	0
$901$	4895.56	0.181015
$902$	−9922.94	−0.366295
$903$	0	0
$904$	13205.8	0.485862
$905$	−14604.2	−0.536418
$906$	0	0
$907$	9279.91	0.339729	0.169865	−	0.985467i	$-0.445667\pi$
$907$	9279.91	0.339729	0.169865	+	0.985467i	$0.445667\pi$
$908$	11197.4	0.409248
$909$	0	0
$910$	−10701.4	−0.389832
$911$	16400.6	0.596461	0.298231	−	0.954494i	$-0.403604\pi$
$911$	16400.6	0.596461	0.298231	+	0.954494i	$0.403604\pi$
$912$	0	0
$913$	17490.4	0.634008
$914$	15782.3	0.571150
$915$	0	0
$916$	5651.38	0.203850
$917$	725.848	0.0261392
$918$	0	0
$919$	−11704.6	−0.420128	−0.210064	−	0.977688i	$-0.567367\pi$
$919$	−11704.6	−0.420128	−0.210064	+	0.977688i	$0.567367\pi$
$920$	7927.57	0.284092
$921$	0	0
$922$	6524.37	0.233046
$923$	−29717.3	−1.05976
$924$	0	0
$925$	93412.7	3.32042
$926$	−7380.81	−0.261931
$927$	0	0
$928$	−349.025	−0.0123462
$929$	16324.4	0.576520	0.288260	−	0.957552i	$-0.406923\pi$
$929$	16324.4	0.576520	0.288260	+	0.957552i	$0.406923\pi$
$930$	0	0
$931$	3194.03	0.112438
$932$	8135.25	0.285922
$933$	0	0
$934$	20386.4	0.714201
$935$	−6282.36	−0.219738
$936$	0	0
$937$	−7397.79	−0.257925	−0.128962	−	0.991649i	$-0.541165\pi$
$937$	−7397.79	−0.257925	−0.128962	+	0.991649i	$0.541165\pi$
$938$	−4679.37	−0.162886
$939$	0	0
$940$	−37519.8	−1.30187
$941$	7426.88	0.257289	0.128645	−	0.991691i	$-0.458937\pi$
$941$	7426.88	0.257289	0.128645	+	0.991691i	$0.458937\pi$
$942$	0	0
$943$	13202.7	0.455928
$944$	852.146	0.0293803
$945$	0	0
$946$	15438.3	0.530596
$947$	−23051.9	−0.791011	−0.395505	−	0.918464i	$-0.629430\pi$
$947$	−23051.9	−0.791011	−0.395505	+	0.918464i	$0.629430\pi$
$948$	0	0
$949$	−21938.9	−0.750438
$950$	5586.71	0.190797
$951$	0	0
$952$	792.293	0.0269731
$953$	−23445.7	−0.796936	−0.398468	−	0.917182i	$-0.630458\pi$
$953$	−23445.7	−0.796936	−0.398468	+	0.917182i	$0.630458\pi$
$954$	0	0
$955$	−71886.4	−2.43580
$956$	−2086.45	−0.0705863
$957$	0	0
$958$	−1680.14	−0.0566626
$959$	−665.739	−0.0224169
$960$	0	0
$961$	−6805.02	−0.228425
$962$	−31759.4	−1.06441
$963$	0	0
$964$	23662.6	0.790582
$965$	−86260.0	−2.87752
$966$	0	0
$967$	−16741.9	−0.556756	−0.278378	−	0.960472i	$-0.589797\pi$
$967$	−16741.9	−0.556756	−0.278378	+	0.960472i	$0.589797\pi$
$968$	7838.24	0.260259
$969$	0	0
$970$	10894.9	0.360634
$971$	−43037.7	−1.42240	−0.711198	−	0.702992i	$-0.751847\pi$
$971$	−43037.7	−1.42240	−0.711198	+	0.702992i	$0.751847\pi$
$972$	0	0
$973$	2310.59	0.0761295
$974$	6734.57	0.221550
$975$	0	0
$976$	−4703.10	−0.154244
$977$	−17899.8	−0.586147	−0.293074	−	0.956090i	$-0.594678\pi$
$977$	−17899.8	−0.586147	−0.293074	+	0.956090i	$0.594678\pi$
$978$	0	0
$979$	6911.95	0.225645
$980$	24523.1	0.799350
$981$	0	0
$982$	37738.2	1.22635
$983$	−46300.3	−1.50229	−0.751144	−	0.660138i	$-0.770498\pi$
$983$	−46300.3	−1.50229	−0.751144	+	0.660138i	$0.770498\pi$
$984$	0	0
$985$	65298.2	2.11226
$986$	−368.012	−0.0118863
$987$	0	0
$988$	−1899.43	−0.0611628
$989$	−20541.1	−0.660435
$990$	0	0
$991$	4621.74	0.148148	0.0740739	−	0.997253i	$-0.476400\pi$
$991$	4621.74	0.148148	0.0740739	+	0.997253i	$0.476400\pi$
$992$	−4851.56	−0.155279
$993$	0	0
$994$	−7606.35	−0.242715
$995$	−6602.25	−0.210357
$996$	0	0
$997$	19944.9	0.633561	0.316780	−	0.948499i	$-0.397398\pi$
$997$	19944.9	0.633561	0.316780	+	0.948499i	$0.397398\pi$
$998$	29132.0	0.924006
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.a.f.1.1		2
3.2	odd	2		162.4.a.g.1.2		2
4.3	odd	2		1296.4.a.l.1.1		2
9.2	odd	6		54.4.c.b.37.1		4
9.4	even	3		18.4.c.b.7.1	✓	4
9.5	odd	6		54.4.c.b.19.1		4
9.7	even	3		18.4.c.b.13.1	yes	4
12.11	even	2		1296.4.a.r.1.2		2
36.7	odd	6		144.4.i.b.49.2		4
36.11	even	6		432.4.i.b.145.1		4
36.23	even	6		432.4.i.b.289.1		4
36.31	odd	6		144.4.i.b.97.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.4.c.b.7.1	✓	4	9.4	even	3
18.4.c.b.13.1	yes	4	9.7	even	3
54.4.c.b.19.1		4	9.5	odd	6
54.4.c.b.37.1		4	9.2	odd	6
144.4.i.b.49.2		4	36.7	odd	6
144.4.i.b.97.2		4	36.31	odd	6
162.4.a.f.1.1		2	1.1	even	1	trivial
162.4.a.g.1.2		2	3.2	odd	2
432.4.i.b.145.1		4	36.11	even	6
432.4.i.b.289.1		4	36.23	even	6
1296.4.a.l.1.1		2	4.3	odd	2
1296.4.a.r.1.2		2	12.11	even	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -19.8704 q^{5} -5.87043 q^{7} -8.00000 q^{8} +39.7409 q^{10} +18.7409 q^{11} +45.8704 q^{13} +11.7409 q^{14} +16.0000 q^{16} +16.8704 q^{17} -10.3521 q^{19} -79.4817 q^{20} -37.4817 q^{22} +49.8704 q^{23} +269.834 q^{25} -91.7409 q^{26} -23.4817 q^{28} +10.9070 q^{29} +151.611 q^{31} -32.0000 q^{32} -33.7409 q^{34} +116.648 q^{35} +346.186 q^{37} +20.7043 q^{38} +158.963 q^{40} +264.741 q^{41} -411.890 q^{43} +74.9634 q^{44} -99.7409 q^{46} +472.056 q^{47} -308.538 q^{49} -539.668 q^{50} +183.482 q^{52} +290.186 q^{53} -372.389 q^{55} +46.9634 q^{56} -21.8140 q^{58} +53.2591 q^{59} -293.944 q^{61} -303.223 q^{62} +64.0000 q^{64} -911.465 q^{65} -398.555 q^{67} +67.4817 q^{68} -233.296 q^{70} -647.854 q^{71} -478.279 q^{73} -692.372 q^{74} -41.4085 q^{76} -110.017 q^{77} +374.316 q^{79} -317.927 q^{80} -529.482 q^{82} +933.279 q^{83} -335.223 q^{85} +823.780 q^{86} -149.927 q^{88} +368.817 q^{89} -269.279 q^{91} +199.482 q^{92} -944.113 q^{94} +205.701 q^{95} +274.149 q^{97} +617.076 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} - 9 q^{5} + 19 q^{7} - 16 q^{8} + 18 q^{10} - 24 q^{11} + 61 q^{13} - 38 q^{14} + 32 q^{16} + 3 q^{17} + 133 q^{19} - 36 q^{20} + 48 q^{22} + 69 q^{23} + 263 q^{25} - 122 q^{26}+ \cdots + 66 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 - 9 * q^5 + 19 * q^7 - 16 * q^8 + 18 * q^10 - 24 * q^11 + 61 * q^13 - 38 * q^14 + 32 * q^16 + 3 * q^17 + 133 * q^19 - 36 * q^20 + 48 * q^22 + 69 * q^23 + 263 * q^25 - 122 * q^26 + 76 * q^28 + 237 * q^29 + 211 * q^31 - 64 * q^32 - 6 * q^34 + 387 * q^35 + 262 * q^37 - 266 * q^38 + 72 * q^40 + 468 * q^41 - 86 * q^43 - 96 * q^44 - 138 * q^46 + 483 * q^47 - 33 * q^49 - 526 * q^50 + 244 * q^52 + 150 * q^53 - 837 * q^55 - 152 * q^56 - 474 * q^58 + 168 * q^59 - 1049 * q^61 - 422 * q^62 + 128 * q^64 - 747 * q^65 - 1166 * q^67 + 12 * q^68 - 774 * q^70 - 312 * q^71 - 311 * q^73 - 524 * q^74 + 532 * q^76 - 1173 * q^77 + 349 * q^79 - 144 * q^80 - 936 * q^82 + 1221 * q^83 - 486 * q^85 + 172 * q^86 + 192 * q^88 - 492 * q^89 + 107 * q^91 + 276 * q^92 - 966 * q^94 + 1764 * q^95 - 128 * q^97 + 66 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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