LMFDB - Embedded newform 189.4.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,4,Mod(1,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("189.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	189.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:	$11.1513609911$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	189.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.73205 q^{2} +14.3923 q^{4} +9.92820 q^{5} +7.00000 q^{7} +30.2487 q^{8} +O(q^{10})$	\(q+4.73205 q^{2} +14.3923 q^{4} +9.92820 q^{5} +7.00000 q^{7} +30.2487 q^{8} +46.9808 q^{10} -3.71281 q^{11} -15.5692 q^{13} +33.1244 q^{14} +28.0000 q^{16} -33.4974 q^{17} +135.923 q^{19} +142.890 q^{20} -17.5692 q^{22} -87.7795 q^{23} -26.4308 q^{25} -73.6743 q^{26} +100.746 q^{28} +242.354 q^{29} -194.708 q^{31} -109.492 q^{32} -158.512 q^{34} +69.4974 q^{35} -239.708 q^{37} +643.195 q^{38} +300.315 q^{40} +470.338 q^{41} -448.215 q^{43} -53.4359 q^{44} -415.377 q^{46} +4.15906 q^{47} +49.0000 q^{49} -125.072 q^{50} -224.077 q^{52} -736.543 q^{53} -36.8616 q^{55} +211.741 q^{56} +1146.83 q^{58} +279.564 q^{59} -514.831 q^{61} -921.367 q^{62} -742.123 q^{64} -154.574 q^{65} -102.123 q^{67} -482.105 q^{68} +328.865 q^{70} -44.1539 q^{71} -901.615 q^{73} -1134.31 q^{74} +1956.25 q^{76} -25.9897 q^{77} +1054.31 q^{79} +277.990 q^{80} +2225.67 q^{82} +487.246 q^{83} -332.569 q^{85} -2120.98 q^{86} -112.308 q^{88} +963.682 q^{89} -108.985 q^{91} -1263.35 q^{92} +19.6809 q^{94} +1349.47 q^{95} +726.908 q^{97} +231.870 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{2} + 8 q^{4} + 6 q^{5} + 14 q^{7} + 12 q^{8}+O(q^{10}) $ 2 * q + 6 * q^2 + 8 * q^4 + 6 * q^5 + 14 * q^7 + 12 * q^8	$ 2 q + 6 q^{2} + 8 q^{4} + 6 q^{5} + 14 q^{7} + 12 q^{8} + 42 q^{10} + 48 q^{11} + 52 q^{13} + 42 q^{14} + 56 q^{16} + 30 q^{17} + 64 q^{19} + 168 q^{20} + 48 q^{22} + 60 q^{23} - 136 q^{25} + 12 q^{26} + 56 q^{28} + 360 q^{29} - 140 q^{31} + 72 q^{32} - 78 q^{34} + 42 q^{35} - 230 q^{37} + 552 q^{38} + 372 q^{40} + 234 q^{41} - 938 q^{43} - 384 q^{44} - 228 q^{46} + 618 q^{47} + 98 q^{49} - 264 q^{50} - 656 q^{52} - 420 q^{53} - 240 q^{55} + 84 q^{56} + 1296 q^{58} + 282 q^{59} - 32 q^{61} - 852 q^{62} - 736 q^{64} - 420 q^{65} + 544 q^{67} - 888 q^{68} + 294 q^{70} - 504 q^{71} - 764 q^{73} - 1122 q^{74} + 2416 q^{76} + 336 q^{77} + 238 q^{79} + 168 q^{80} + 1926 q^{82} - 522 q^{83} - 582 q^{85} - 2742 q^{86} - 1056 q^{88} + 708 q^{89} + 364 q^{91} - 2208 q^{92} + 798 q^{94} + 1632 q^{95} + 664 q^{97} + 294 q^{98}+O(q^{100}) $ 2 * q + 6 * q^2 + 8 * q^4 + 6 * q^5 + 14 * q^7 + 12 * q^8 + 42 * q^10 + 48 * q^11 + 52 * q^13 + 42 * q^14 + 56 * q^16 + 30 * q^17 + 64 * q^19 + 168 * q^20 + 48 * q^22 + 60 * q^23 - 136 * q^25 + 12 * q^26 + 56 * q^28 + 360 * q^29 - 140 * q^31 + 72 * q^32 - 78 * q^34 + 42 * q^35 - 230 * q^37 + 552 * q^38 + 372 * q^40 + 234 * q^41 - 938 * q^43 - 384 * q^44 - 228 * q^46 + 618 * q^47 + 98 * q^49 - 264 * q^50 - 656 * q^52 - 420 * q^53 - 240 * q^55 + 84 * q^56 + 1296 * q^58 + 282 * q^59 - 32 * q^61 - 852 * q^62 - 736 * q^64 - 420 * q^65 + 544 * q^67 - 888 * q^68 + 294 * q^70 - 504 * q^71 - 764 * q^73 - 1122 * q^74 + 2416 * q^76 + 336 * q^77 + 238 * q^79 + 168 * q^80 + 1926 * q^82 - 522 * q^83 - 582 * q^85 - 2742 * q^86 - 1056 * q^88 + 708 * q^89 + 364 * q^91 - 2208 * q^92 + 798 * q^94 + 1632 * q^95 + 664 * q^97 + 294 * 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 $

$2$	4.73205	1.67303	0.836516	−	0.547942i	$-0.184589\pi$
$2$	4.73205	1.67303	0.836516	+	0.547942i	$0.184589\pi$
$3$	0	0
$3$	0	0
$4$	14.3923	1.79904
$5$	9.92820	0.888005	0.444003	−	0.896025i	$-0.353558\pi$
$5$	9.92820	0.888005	0.444003	+	0.896025i	$0.353558\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	30.2487	1.33682
$9$	0	0
$10$	46.9808	1.48566
$11$	−3.71281	−0.101769	−0.0508843	−	0.998705i	$-0.516204\pi$
$11$	−3.71281	−0.101769	−0.0508843	+	0.998705i	$0.516204\pi$
$12$	0	0
$13$	−15.5692	−0.332163	−0.166082	−	0.986112i	$-0.553112\pi$
$13$	−15.5692	−0.332163	−0.166082	+	0.986112i	$0.553112\pi$
$14$	33.1244	0.632347
$15$	0	0
$16$	28.0000	0.437500
$17$	−33.4974	−0.477901	−0.238951	−	0.971032i	$-0.576803\pi$
$17$	−33.4974	−0.477901	−0.238951	+	0.971032i	$0.576803\pi$
$18$	0	0
$19$	135.923	1.64120	0.820602	−	0.571500i	$-0.193638\pi$
$19$	135.923	1.64120	0.820602	+	0.571500i	$0.193638\pi$
$20$	142.890	1.59756
$21$	0	0
$22$	−17.5692	−0.170262
$23$	−87.7795	−0.795795	−0.397897	−	0.917430i	$-0.630260\pi$
$23$	−87.7795	−0.795795	−0.397897	+	0.917430i	$0.630260\pi$
$24$	0	0
$25$	−26.4308	−0.211446
$26$	−73.6743	−0.555720
$27$	0	0
$28$	100.746	0.679972
$29$	242.354	1.55186	0.775931	−	0.630818i	$-0.217281\pi$
$29$	242.354	1.55186	0.775931	+	0.630818i	$0.217281\pi$
$30$	0	0
$31$	−194.708	−1.12808	−0.564041	−	0.825747i	$-0.690754\pi$
$31$	−194.708	−1.12808	−0.564041	+	0.825747i	$0.690754\pi$
$32$	−109.492	−0.604865
$33$	0	0
$34$	−158.512	−0.799544
$35$	69.4974	0.335635
$36$	0	0
$37$	−239.708	−1.06507	−0.532536	−	0.846407i	$-0.678761\pi$
$37$	−239.708	−1.06507	−0.532536	+	0.846407i	$0.678761\pi$
$38$	643.195	2.74579
$39$	0	0
$40$	300.315	1.18710
$41$	470.338	1.79157	0.895787	−	0.444484i	$-0.146613\pi$
$41$	470.338	1.79157	0.895787	+	0.444484i	$0.146613\pi$
$42$	0	0
$43$	−448.215	−1.58959	−0.794793	−	0.606880i	$-0.792421\pi$
$43$	−448.215	−1.58959	−0.794793	+	0.606880i	$0.792421\pi$
$44$	−53.4359	−0.183086
$45$	0	0
$46$	−415.377	−1.33139
$47$	4.15906	0.0129077	0.00645384	−	0.999979i	$-0.497946\pi$
$47$	4.15906	0.0129077	0.00645384	+	0.999979i	$0.497946\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−125.072	−0.353756
$51$	0	0
$52$	−224.077	−0.597575
$53$	−736.543	−1.90891	−0.954453	−	0.298361i	$-0.903560\pi$
$53$	−736.543	−1.90891	−0.954453	+	0.298361i	$0.903560\pi$
$54$	0	0
$55$	−36.8616	−0.0903711
$56$	211.741	0.505269
$57$	0	0
$58$	1146.83	2.59631
$59$	279.564	0.616884	0.308442	−	0.951243i	$-0.400192\pi$
$59$	279.564	0.616884	0.308442	+	0.951243i	$0.400192\pi$
$60$	0	0
$61$	−514.831	−1.08061	−0.540306	−	0.841469i	$-0.681691\pi$
$61$	−514.831	−1.08061	−0.540306	+	0.841469i	$0.681691\pi$
$62$	−921.367	−1.88732
$63$	0	0
$64$	−742.123	−1.44946
$65$	−154.574	−0.294963
$66$	0	0
$67$	−102.123	−0.186214	−0.0931068	−	0.995656i	$-0.529680\pi$
$67$	−102.123	−0.186214	−0.0931068	+	0.995656i	$0.529680\pi$
$68$	−482.105	−0.859762
$69$	0	0
$70$	328.865	0.561528
$71$	−44.1539	−0.0738043	−0.0369021	−	0.999319i	$-0.511749\pi$
$71$	−44.1539	−0.0738043	−0.0369021	+	0.999319i	$0.511749\pi$
$72$	0	0
$73$	−901.615	−1.44556	−0.722781	−	0.691077i	$-0.757137\pi$
$73$	−901.615	−1.44556	−0.722781	+	0.691077i	$0.757137\pi$
$74$	−1134.31	−1.78190
$75$	0	0
$76$	1956.25	2.95259
$77$	−25.9897	−0.0384649
$78$	0	0
$79$	1054.31	1.50150	0.750752	−	0.660584i	$-0.229691\pi$
$79$	1054.31	1.50150	0.750752	+	0.660584i	$0.229691\pi$
$80$	277.990	0.388502
$81$	0	0
$82$	2225.67	2.99736
$83$	487.246	0.644364	0.322182	−	0.946678i	$-0.395584\pi$
$83$	487.246	0.644364	0.322182	+	0.946678i	$0.395584\pi$
$84$	0	0
$85$	−332.569	−0.424379
$86$	−2120.98	−2.65943
$87$	0	0
$88$	−112.308	−0.136046
$89$	963.682	1.14775	0.573877	−	0.818942i	$-0.305439\pi$
$89$	963.682	1.14775	0.573877	+	0.818942i	$0.305439\pi$
$90$	0	0
$91$	−108.985	−0.125546
$92$	−1263.35	−1.43167
$93$	0	0
$94$	19.6809	0.0215950
$95$	1349.47	1.45740
$96$	0	0
$97$	726.908	0.760890	0.380445	−	0.924804i	$-0.375771\pi$
$97$	726.908	0.760890	0.380445	+	0.924804i	$0.375771\pi$
$98$	231.870	0.239005
$99$	0	0
$100$	−380.400	−0.380400
$101$	268.872	0.264889	0.132444	−	0.991190i	$-0.457717\pi$
$101$	268.872	0.264889	0.132444	+	0.991190i	$0.457717\pi$
$102$	0	0
$103$	1108.35	1.06028	0.530142	−	0.847909i	$-0.322138\pi$
$103$	1108.35	1.06028	0.530142	+	0.847909i	$0.322138\pi$
$104$	−470.949	−0.444042
$105$	0	0
$106$	−3485.36	−3.19366
$107$	84.4665	0.0763148	0.0381574	−	0.999272i	$-0.487851\pi$
$107$	84.4665	0.0763148	0.0381574	+	0.999272i	$0.487851\pi$
$108$	0	0
$109$	−174.169	−0.153050	−0.0765248	−	0.997068i	$-0.524382\pi$
$109$	−174.169	−0.153050	−0.0765248	+	0.997068i	$0.524382\pi$
$110$	−174.431	−0.151194
$111$	0	0
$112$	196.000	0.165359
$113$	1049.77	0.873929	0.436964	−	0.899479i	$-0.356054\pi$
$113$	1049.77	0.873929	0.436964	+	0.899479i	$0.356054\pi$
$114$	0	0
$115$	−871.492	−0.706670
$116$	3488.03	2.79186
$117$	0	0
$118$	1322.91	1.03207
$119$	−234.482	−0.180630
$120$	0	0
$121$	−1317.22	−0.989643
$122$	−2436.20	−1.80790
$123$	0	0
$124$	−2802.29	−2.02946
$125$	−1503.44	−1.07577
$126$	0	0
$127$	−293.969	−0.205398	−0.102699	−	0.994712i	$-0.532748\pi$
$127$	−293.969	−0.205398	−0.102699	+	0.994712i	$0.532748\pi$
$128$	−2635.83	−1.82013
$129$	0	0
$130$	−731.454	−0.493483
$131$	1110.76	0.740820	0.370410	−	0.928868i	$-0.379217\pi$
$131$	1110.76	0.740820	0.370410	+	0.928868i	$0.379217\pi$
$132$	0	0
$133$	951.461	0.620317
$134$	−483.251	−0.311541
$135$	0	0
$136$	−1013.25	−0.638866
$137$	−238.113	−0.148492	−0.0742458	−	0.997240i	$-0.523655\pi$
$137$	−238.113	−0.148492	−0.0742458	+	0.997240i	$0.523655\pi$
$138$	0	0
$139$	−189.108	−0.115395	−0.0576974	−	0.998334i	$-0.518376\pi$
$139$	−189.108	−0.115395	−0.0576974	+	0.998334i	$0.518376\pi$
$140$	1000.23	0.603819
$141$	0	0
$142$	−208.939	−0.123477
$143$	57.8056	0.0338038
$144$	0	0
$145$	2406.14	1.37806
$146$	−4266.49	−2.41847
$147$	0	0
$148$	−3449.95	−1.91611
$149$	1107.59	0.608975	0.304487	−	0.952516i	$-0.401515\pi$
$149$	1107.59	0.608975	0.304487	+	0.952516i	$0.401515\pi$
$150$	0	0
$151$	3512.55	1.89303	0.946515	−	0.322660i	$-0.104577\pi$
$151$	3512.55	1.89303	0.946515	+	0.322660i	$0.104577\pi$
$152$	4111.50	2.19399
$153$	0	0
$154$	−122.985	−0.0643531
$155$	−1933.10	−1.00174
$156$	0	0
$157$	524.169	0.266454	0.133227	−	0.991086i	$-0.457466\pi$
$157$	524.169	0.266454	0.133227	+	0.991086i	$0.457466\pi$
$158$	4989.04	2.51207
$159$	0	0
$160$	−1087.06	−0.537123
$161$	−614.456	−0.300782
$162$	0	0
$163$	245.693	0.118062	0.0590311	−	0.998256i	$-0.481199\pi$
$163$	245.693	0.118062	0.0590311	+	0.998256i	$0.481199\pi$
$164$	6769.25	3.22311
$165$	0	0
$166$	2305.67	1.07804
$167$	586.785	0.271897	0.135948	−	0.990716i	$-0.456592\pi$
$167$	586.785	0.271897	0.135948	+	0.990716i	$0.456592\pi$
$168$	0	0
$169$	−1954.60	−0.889667
$170$	−1573.73	−0.710000
$171$	0	0
$172$	−6450.85	−2.85973
$173$	2957.80	1.29987	0.649936	−	0.759989i	$-0.274796\pi$
$173$	2957.80	1.29987	0.649936	+	0.759989i	$0.274796\pi$
$174$	0	0
$175$	−185.015	−0.0799192
$176$	−103.959	−0.0445238
$177$	0	0
$178$	4560.19	1.92023
$179$	2724.73	1.13774	0.568872	−	0.822426i	$-0.307380\pi$
$179$	2724.73	1.13774	0.568872	+	0.822426i	$0.307380\pi$
$180$	0	0
$181$	446.138	0.183211	0.0916055	−	0.995795i	$-0.470800\pi$
$181$	446.138	0.183211	0.0916055	+	0.995795i	$0.470800\pi$
$182$	−515.720	−0.210043
$183$	0	0
$184$	−2655.22	−1.06383
$185$	−2379.87	−0.945791
$186$	0	0
$187$	124.370	0.0486354
$188$	59.8584	0.0232214
$189$	0	0
$190$	6385.77	2.43828
$191$	3517.90	1.33270	0.666351	−	0.745638i	$-0.267855\pi$
$191$	3517.90	1.33270	0.666351	+	0.745638i	$0.267855\pi$
$192$	0	0
$193$	4869.61	1.81618	0.908090	−	0.418776i	$-0.137541\pi$
$193$	4869.61	1.81618	0.908090	+	0.418776i	$0.137541\pi$
$194$	3439.76	1.27299
$195$	0	0
$196$	705.223	0.257005
$197$	−3427.46	−1.23958	−0.619788	−	0.784769i	$-0.712781\pi$
$197$	−3427.46	−1.23958	−0.619788	+	0.784769i	$0.712781\pi$
$198$	0	0
$199$	−4012.51	−1.42934	−0.714671	−	0.699461i	$-0.753424\pi$
$199$	−4012.51	−1.42934	−0.714671	+	0.699461i	$0.753424\pi$
$200$	−799.497	−0.282665
$201$	0	0
$202$	1272.32	0.443167
$203$	1696.48	0.586548
$204$	0	0
$205$	4669.61	1.59093
$206$	5244.79	1.77389
$207$	0	0
$208$	−435.938	−0.145321
$209$	−504.657	−0.167023
$210$	0	0
$211$	4281.08	1.39678	0.698392	−	0.715715i	$-0.253899\pi$
$211$	4281.08	1.39678	0.698392	+	0.715715i	$0.253899\pi$
$212$	−10600.6	−3.43419
$213$	0	0
$214$	399.700	0.127677
$215$	−4449.97	−1.41156
$216$	0	0
$217$	−1362.95	−0.426375
$218$	−824.178	−0.256057
$219$	0	0
$220$	−530.523	−0.162581
$221$	521.529	0.158741
$222$	0	0
$223$	−764.153	−0.229469	−0.114734	−	0.993396i	$-0.536602\pi$
$223$	−764.153	−0.229469	−0.114734	+	0.993396i	$0.536602\pi$
$224$	−766.446	−0.228617
$225$	0	0
$226$	4967.56	1.46211
$227$	3872.35	1.13223	0.566116	−	0.824325i	$-0.308445\pi$
$227$	3872.35	1.13223	0.566116	+	0.824325i	$0.308445\pi$
$228$	0	0
$229$	−3866.11	−1.11563	−0.557816	−	0.829965i	$-0.688360\pi$
$229$	−3866.11	−1.11563	−0.557816	+	0.829965i	$0.688360\pi$
$230$	−4123.95	−1.18228
$231$	0	0
$232$	7330.89	2.07455
$233$	−5378.29	−1.51220	−0.756102	−	0.654454i	$-0.772898\pi$
$233$	−5378.29	−1.51220	−0.756102	+	0.654454i	$0.772898\pi$
$234$	0	0
$235$	41.2920	0.0114621
$236$	4023.57	1.10980
$237$	0	0
$238$	−1109.58	−0.302199
$239$	4763.15	1.28913	0.644566	−	0.764548i	$-0.277038\pi$
$239$	4763.15	1.28913	0.644566	+	0.764548i	$0.277038\pi$
$240$	0	0
$241$	−318.030	−0.0850047	−0.0425023	−	0.999096i	$-0.513533\pi$
$241$	−318.030	−0.0850047	−0.0425023	+	0.999096i	$0.513533\pi$
$242$	−6233.13	−1.65571
$243$	0	0
$244$	−7409.60	−1.94406
$245$	486.482	0.126858
$246$	0	0
$247$	−2116.22	−0.545148
$248$	−5889.66	−1.50804
$249$	0	0
$250$	−7114.33	−1.79980
$251$	1577.00	0.396571	0.198285	−	0.980144i	$-0.436463\pi$
$251$	1577.00	0.396571	0.198285	+	0.980144i	$0.436463\pi$
$252$	0	0
$253$	325.909	0.0809870
$254$	−1391.08	−0.343638
$255$	0	0
$256$	−6535.88	−1.59567
$257$	−6069.46	−1.47316	−0.736580	−	0.676350i	$-0.763561\pi$
$257$	−6069.46	−1.47316	−0.736580	+	0.676350i	$0.763561\pi$
$258$	0	0
$259$	−1677.95	−0.402560
$260$	−2224.68	−0.530650
$261$	0	0
$262$	5256.17	1.23942
$263$	−3654.73	−0.856883	−0.428441	−	0.903570i	$-0.640937\pi$
$263$	−3654.73	−0.856883	−0.428441	+	0.903570i	$0.640937\pi$
$264$	0	0
$265$	−7312.55	−1.69512
$266$	4502.36	1.03781
$267$	0	0
$268$	−1469.78	−0.335005
$269$	−435.652	−0.0987441	−0.0493721	−	0.998780i	$-0.515722\pi$
$269$	−435.652	−0.0987441	−0.0493721	+	0.998780i	$0.515722\pi$
$270$	0	0
$271$	1230.98	0.275930	0.137965	−	0.990437i	$-0.455944\pi$
$271$	1230.98	0.275930	0.137965	+	0.990437i	$0.455944\pi$
$272$	−937.928	−0.209082
$273$	0	0
$274$	−1126.76	−0.248431
$275$	98.1325	0.0215186
$276$	0	0
$277$	3994.17	0.866377	0.433188	−	0.901303i	$-0.357388\pi$
$277$	3994.17	0.866377	0.433188	+	0.901303i	$0.357388\pi$
$278$	−894.866	−0.193059
$279$	0	0
$280$	2102.21	0.448682
$281$	−3615.71	−0.767598	−0.383799	−	0.923417i	$-0.625384\pi$
$281$	−3615.71	−0.767598	−0.383799	+	0.923417i	$0.625384\pi$
$282$	0	0
$283$	−2353.42	−0.494332	−0.247166	−	0.968973i	$-0.579499\pi$
$283$	−2353.42	−0.494332	−0.247166	+	0.968973i	$0.579499\pi$
$284$	−635.476	−0.132777
$285$	0	0
$286$	273.539	0.0565549
$287$	3292.37	0.677151
$288$	0	0
$289$	−3790.92	−0.771611
$290$	11386.0	2.30554
$291$	0	0
$292$	−12976.3	−2.60062
$293$	3580.68	0.713943	0.356972	−	0.934115i	$-0.383809\pi$
$293$	3580.68	0.713943	0.356972	+	0.934115i	$0.383809\pi$
$294$	0	0
$295$	2775.57	0.547796
$296$	−7250.85	−1.42381
$297$	0	0
$298$	5241.17	1.01883
$299$	1366.66	0.264334
$300$	0	0
$301$	−3137.51	−0.600807
$302$	16621.6	3.16710
$303$	0	0
$304$	3805.85	0.718027
$305$	−5111.34	−0.959589
$306$	0	0
$307$	−9077.92	−1.68764	−0.843818	−	0.536629i	$-0.819697\pi$
$307$	−9077.92	−1.68764	−0.843818	+	0.536629i	$0.819697\pi$
$308$	−374.052	−0.0691999
$309$	0	0
$310$	−9147.51	−1.67595
$311$	−8966.68	−1.63490	−0.817450	−	0.576000i	$-0.804613\pi$
$311$	−8966.68	−1.63490	−0.817450	+	0.576000i	$0.804613\pi$
$312$	0	0
$313$	7191.74	1.29873	0.649363	−	0.760479i	$-0.275036\pi$
$313$	7191.74	1.29873	0.649363	+	0.760479i	$0.275036\pi$
$314$	2480.39	0.445786
$315$	0	0
$316$	15173.9	2.70126
$317$	−2288.95	−0.405553	−0.202776	−	0.979225i	$-0.564996\pi$
$317$	−2288.95	−0.405553	−0.202776	+	0.979225i	$0.564996\pi$
$318$	0	0
$319$	−899.814	−0.157931
$320$	−7367.95	−1.28713
$321$	0	0
$322$	−2907.64	−0.503218
$323$	−4553.07	−0.784333
$324$	0	0
$325$	411.507	0.0702347
$326$	1162.63	0.197522
$327$	0	0
$328$	14227.1	2.39501
$329$	29.1134	0.00487864
$330$	0	0
$331$	−6298.61	−1.04593	−0.522965	−	0.852354i	$-0.675174\pi$
$331$	−6298.61	−1.04593	−0.522965	+	0.852354i	$0.675174\pi$
$332$	7012.59	1.15923
$333$	0	0
$334$	2776.70	0.454892
$335$	−1013.90	−0.165359
$336$	0	0
$337$	189.615	0.0306498	0.0153249	−	0.999883i	$-0.495122\pi$
$337$	189.615	0.0306498	0.0153249	+	0.999883i	$0.495122\pi$
$338$	−9249.26	−1.48844
$339$	0	0
$340$	−4786.44	−0.763474
$341$	722.913	0.114803
$342$	0	0
$343$	343.000	0.0539949
$344$	−13557.9	−2.12499
$345$	0	0
$346$	13996.5	2.17473
$347$	50.4728	0.00780841	0.00390421	−	0.999992i	$-0.498757\pi$
$347$	50.4728	0.00780841	0.00390421	+	0.999992i	$0.498757\pi$
$348$	0	0
$349$	−1338.71	−0.205328	−0.102664	−	0.994716i	$-0.532737\pi$
$349$	−1338.71	−0.205328	−0.102664	+	0.994716i	$0.532737\pi$
$350$	−875.503	−0.133707
$351$	0	0
$352$	406.524	0.0615563
$353$	3192.99	0.481433	0.240717	−	0.970595i	$-0.422618\pi$
$353$	3192.99	0.481433	0.240717	+	0.970595i	$0.422618\pi$
$354$	0	0
$355$	−438.369	−0.0655386
$356$	13869.6	2.06485
$357$	0	0
$358$	12893.6	1.90348
$359$	6661.22	0.979292	0.489646	−	0.871921i	$-0.337126\pi$
$359$	6661.22	0.979292	0.489646	+	0.871921i	$0.337126\pi$
$360$	0	0
$361$	11616.1	1.69355
$362$	2111.15	0.306518
$363$	0	0
$364$	−1568.54	−0.225862
$365$	−8951.42	−1.28367
$366$	0	0
$367$	4592.77	0.653244	0.326622	−	0.945155i	$-0.394090\pi$
$367$	4592.77	0.653244	0.326622	+	0.945155i	$0.394090\pi$
$368$	−2457.82	−0.348160
$369$	0	0
$370$	−11261.6	−1.58234
$371$	−5155.80	−0.721499
$372$	0	0
$373$	6982.57	0.969286	0.484643	−	0.874712i	$-0.338950\pi$
$373$	6982.57	0.969286	0.484643	+	0.874712i	$0.338950\pi$
$374$	588.524	0.0813685
$375$	0	0
$376$	125.806	0.0172552
$377$	−3773.26	−0.515472
$378$	0	0
$379$	−12060.4	−1.63457	−0.817286	−	0.576233i	$-0.804522\pi$
$379$	−12060.4	−1.63457	−0.817286	+	0.576233i	$0.804522\pi$
$380$	19422.0	2.62192
$381$	0	0
$382$	16646.9	2.22965
$383$	9674.46	1.29071	0.645355	−	0.763883i	$-0.276709\pi$
$383$	9674.46	1.29071	0.645355	+	0.763883i	$0.276709\pi$
$384$	0	0
$385$	−258.031	−0.0341571
$386$	23043.3	3.03853
$387$	0	0
$388$	10461.9	1.36887
$389$	−11377.9	−1.48299	−0.741496	−	0.670957i	$-0.765883\pi$
$389$	−11377.9	−1.48299	−0.741496	+	0.670957i	$0.765883\pi$
$390$	0	0
$391$	2940.39	0.380311
$392$	1482.19	0.190974
$393$	0	0
$394$	−16218.9	−2.07385
$395$	10467.4	1.33334
$396$	0	0
$397$	−15202.7	−1.92192	−0.960958	−	0.276693i	$-0.910762\pi$
$397$	−15202.7	−1.92192	−0.960958	+	0.276693i	$0.910762\pi$
$398$	−18987.4	−2.39134
$399$	0	0
$400$	−740.062	−0.0925077
$401$	−1396.02	−0.173851	−0.0869254	−	0.996215i	$-0.527704\pi$
$401$	−1396.02	−0.173851	−0.0869254	+	0.996215i	$0.527704\pi$
$402$	0	0
$403$	3031.45	0.374707
$404$	3869.69	0.476545
$405$	0	0
$406$	8027.81	0.981315
$407$	889.990	0.108391
$408$	0	0
$409$	−8673.54	−1.04860	−0.524302	−	0.851533i	$-0.675674\pi$
$409$	−8673.54	−1.04860	−0.524302	+	0.851533i	$0.675674\pi$
$410$	22096.9	2.66167
$411$	0	0
$412$	15951.8	1.90749
$413$	1956.95	0.233160
$414$	0	0
$415$	4837.48	0.572199
$416$	1704.71	0.200914
$417$	0	0
$418$	−2388.06	−0.279435
$419$	7313.92	0.852765	0.426382	−	0.904543i	$-0.359788\pi$
$419$	7313.92	0.852765	0.426382	+	0.904543i	$0.359788\pi$
$420$	0	0
$421$	−7695.26	−0.890841	−0.445420	−	0.895322i	$-0.646946\pi$
$421$	−7695.26	−0.890841	−0.445420	+	0.895322i	$0.646946\pi$
$422$	20258.3	2.33687
$423$	0	0
$424$	−22279.5	−2.55186
$425$	885.363	0.101050
$426$	0	0
$427$	−3603.81	−0.408433
$428$	1215.67	0.137293
$429$	0	0
$430$	−21057.5	−2.36159
$431$	6258.10	0.699402	0.349701	−	0.936861i	$-0.386283\pi$
$431$	6258.10	0.699402	0.349701	+	0.936861i	$0.386283\pi$
$432$	0	0
$433$	−11923.3	−1.32332	−0.661660	−	0.749804i	$-0.730148\pi$
$433$	−11923.3	−1.32332	−0.661660	+	0.749804i	$0.730148\pi$
$434$	−6449.57	−0.713339
$435$	0	0
$436$	−2506.70	−0.275342
$437$	−11931.3	−1.30606
$438$	0	0
$439$	17115.6	1.86079	0.930393	−	0.366564i	$-0.119466\pi$
$439$	17115.6	1.86079	0.930393	+	0.366564i	$0.119466\pi$
$440$	−1115.01	−0.120810
$441$	0	0
$442$	2467.90	0.265579
$443$	−8095.04	−0.868187	−0.434094	−	0.900868i	$-0.642931\pi$
$443$	−8095.04	−0.868187	−0.434094	+	0.900868i	$0.642931\pi$
$444$	0	0
$445$	9567.63	1.01921
$446$	−3616.01	−0.383908
$447$	0	0
$448$	−5194.86	−0.547844
$449$	−6156.71	−0.647111	−0.323556	−	0.946209i	$-0.604878\pi$
$449$	−6156.71	−0.647111	−0.323556	+	0.946209i	$0.604878\pi$
$450$	0	0
$451$	−1746.28	−0.182326
$452$	15108.6	1.57223
$453$	0	0
$454$	18324.2	1.89426
$455$	−1082.02	−0.111486
$456$	0	0
$457$	5657.23	0.579068	0.289534	−	0.957168i	$-0.406500\pi$
$457$	5657.23	0.579068	0.289534	+	0.957168i	$0.406500\pi$
$458$	−18294.6	−1.86649
$459$	0	0
$460$	−12542.8	−1.27133
$461$	−1414.61	−0.142918	−0.0714589	−	0.997444i	$-0.522765\pi$
$461$	−1414.61	−0.142918	−0.0714589	+	0.997444i	$0.522765\pi$
$462$	0	0
$463$	−5404.92	−0.542523	−0.271261	−	0.962506i	$-0.587441\pi$
$463$	−5404.92	−0.542523	−0.271261	+	0.962506i	$0.587441\pi$
$464$	6785.91	0.678939
$465$	0	0
$466$	−25450.3	−2.52996
$467$	16952.8	1.67983	0.839916	−	0.542717i	$-0.182604\pi$
$467$	16952.8	1.67983	0.839916	+	0.542717i	$0.182604\pi$
$468$	0	0
$469$	−714.861	−0.0703821
$470$	195.396	0.0191765
$471$	0	0
$472$	8456.45	0.824661
$473$	1664.14	0.161770
$474$	0	0
$475$	−3592.55	−0.347027
$476$	−3374.74	−0.324960
$477$	0	0
$478$	22539.5	2.15676
$479$	−18372.7	−1.75255	−0.876276	−	0.481810i	$-0.839979\pi$
$479$	−18372.7	−1.75255	−0.876276	+	0.481810i	$0.839979\pi$
$480$	0	0
$481$	3732.06	0.353778
$482$	−1504.94	−0.142216
$483$	0	0
$484$	−18957.8	−1.78041
$485$	7216.89	0.675674
$486$	0	0
$487$	−42.5560	−0.00395974	−0.00197987	−	0.999998i	$-0.500630\pi$
$487$	−42.5560	−0.00395974	−0.00197987	+	0.999998i	$0.500630\pi$
$488$	−15573.0	−1.44458
$489$	0	0
$490$	2302.06	0.212237
$491$	−6092.09	−0.559943	−0.279972	−	0.960008i	$-0.590325\pi$
$491$	−6092.09	−0.559943	−0.279972	+	0.960008i	$0.590325\pi$
$492$	0	0
$493$	−8118.23	−0.741636
$494$	−10014.0	−0.912051
$495$	0	0
$496$	−5451.81	−0.493536
$497$	−309.077	−0.0278954
$498$	0	0
$499$	6128.64	0.549811	0.274906	−	0.961471i	$-0.411353\pi$
$499$	6128.64	0.549811	0.274906	+	0.961471i	$0.411353\pi$
$500$	−21637.9	−1.93535
$501$	0	0
$502$	7462.44	0.663476
$503$	−3606.14	−0.319661	−0.159831	−	0.987144i	$-0.551095\pi$
$503$	−3606.14	−0.319661	−0.159831	+	0.987144i	$0.551095\pi$
$504$	0	0
$505$	2669.41	0.235223
$506$	1542.22	0.135494
$507$	0	0
$508$	−4230.90	−0.369519
$509$	−15207.2	−1.32426	−0.662128	−	0.749391i	$-0.730346\pi$
$509$	−15207.2	−1.32426	−0.662128	+	0.749391i	$0.730346\pi$
$510$	0	0
$511$	−6311.31	−0.546371
$512$	−9841.49	−0.849486
$513$	0	0
$514$	−28721.0	−2.46465
$515$	11004.0	0.941539
$516$	0	0
$517$	−15.4418	−0.00131360
$518$	−7940.16	−0.673496
$519$	0	0
$520$	−4675.68	−0.394311
$521$	−5497.74	−0.462304	−0.231152	−	0.972918i	$-0.574250\pi$
$521$	−5497.74	−0.462304	−0.231152	+	0.972918i	$0.574250\pi$
$522$	0	0
$523$	15158.7	1.26739	0.633693	−	0.773584i	$-0.281538\pi$
$523$	15158.7	1.26739	0.633693	+	0.773584i	$0.281538\pi$
$524$	15986.4	1.33276
$525$	0	0
$526$	−17294.4	−1.43359
$527$	6522.20	0.539111
$528$	0	0
$529$	−4461.77	−0.366711
$530$	−34603.4	−2.83599
$531$	0	0
$532$	13693.7	1.11597
$533$	−7322.80	−0.595095
$534$	0	0
$535$	838.601	0.0677680
$536$	−3089.09	−0.248933
$537$	0	0
$538$	−2061.53	−0.165202
$539$	−181.928	−0.0145384
$540$	0	0
$541$	21418.8	1.70216	0.851080	−	0.525037i	$-0.175948\pi$
$541$	21418.8	1.70216	0.851080	+	0.525037i	$0.175948\pi$
$542$	5825.08	0.461639
$543$	0	0
$544$	3667.71	0.289066
$545$	−1729.19	−0.135909
$546$	0	0
$547$	15536.1	1.21439	0.607197	−	0.794551i	$-0.292294\pi$
$547$	15536.1	1.21439	0.607197	+	0.794551i	$0.292294\pi$
$548$	−3426.99	−0.267142
$549$	0	0
$550$	464.368	0.0360013
$551$	32941.5	2.54692
$552$	0	0
$553$	7380.15	0.567515
$554$	18900.6	1.44948
$555$	0	0
$556$	−2721.69	−0.207600
$557$	13975.1	1.06309	0.531547	−	0.847029i	$-0.321611\pi$
$557$	13975.1	1.06309	0.531547	+	0.847029i	$0.321611\pi$
$558$	0	0
$559$	6978.36	0.528002
$560$	1945.93	0.146840
$561$	0	0
$562$	−17109.7	−1.28422
$563$	1117.89	0.0836825	0.0418412	−	0.999124i	$-0.486678\pi$
$563$	1117.89	0.0836825	0.0418412	+	0.999124i	$0.486678\pi$
$564$	0	0
$565$	10422.3	0.776054
$566$	−11136.5	−0.827034
$567$	0	0
$568$	−1335.60	−0.0986628
$569$	14452.1	1.06478	0.532392	−	0.846498i	$-0.321293\pi$
$569$	14452.1	1.06478	0.532392	+	0.846498i	$0.321293\pi$
$570$	0	0
$571$	17396.8	1.27501	0.637506	−	0.770445i	$-0.279966\pi$
$571$	17396.8	1.27501	0.637506	+	0.770445i	$0.279966\pi$
$572$	831.956	0.0608144
$573$	0	0
$574$	15579.7	1.13290
$575$	2320.08	0.168268
$576$	0	0
$577$	12251.2	0.883923	0.441961	−	0.897034i	$-0.354283\pi$
$577$	12251.2	0.883923	0.441961	+	0.897034i	$0.354283\pi$
$578$	−17938.8	−1.29093
$579$	0	0
$580$	34629.9	2.47918
$581$	3410.72	0.243547
$582$	0	0
$583$	2734.65	0.194267
$584$	−27272.7	−1.93245
$585$	0	0
$586$	16943.9	1.19445
$587$	24620.6	1.73118	0.865589	−	0.500755i	$-0.166944\pi$
$587$	24620.6	1.73118	0.865589	+	0.500755i	$0.166944\pi$
$588$	0	0
$589$	−26465.3	−1.85141
$590$	13134.1	0.916481
$591$	0	0
$592$	−6711.81	−0.465969
$593$	−3020.20	−0.209148	−0.104574	−	0.994517i	$-0.533348\pi$
$593$	−3020.20	−0.209148	−0.104574	+	0.994517i	$0.533348\pi$
$594$	0	0
$595$	−2327.98	−0.160400
$596$	15940.8	1.09557
$597$	0	0
$598$	6467.09	0.442239
$599$	−16830.3	−1.14803	−0.574013	−	0.818846i	$-0.694614\pi$
$599$	−16830.3	−1.14803	−0.574013	+	0.818846i	$0.694614\pi$
$600$	0	0
$601$	8783.94	0.596180	0.298090	−	0.954538i	$-0.403650\pi$
$601$	8783.94	0.596180	0.298090	+	0.954538i	$0.403650\pi$
$602$	−14846.8	−1.00517
$603$	0	0
$604$	50553.7	3.40563
$605$	−13077.6	−0.878809
$606$	0	0
$607$	16803.3	1.12360	0.561799	−	0.827274i	$-0.310109\pi$
$607$	16803.3	1.12360	0.561799	+	0.827274i	$0.310109\pi$
$608$	−14882.5	−0.992707
$609$	0	0
$610$	−24187.1	−1.60542
$611$	−64.7533	−0.00428746
$612$	0	0
$613$	1319.88	0.0869647	0.0434823	−	0.999054i	$-0.486155\pi$
$613$	1319.88	0.0869647	0.0434823	+	0.999054i	$0.486155\pi$
$614$	−42957.2	−2.82347
$615$	0	0
$616$	−786.155	−0.0514206
$617$	−2586.39	−0.168759	−0.0843793	−	0.996434i	$-0.526891\pi$
$617$	−2586.39	−0.168759	−0.0843793	+	0.996434i	$0.526891\pi$
$618$	0	0
$619$	5296.63	0.343925	0.171962	−	0.985104i	$-0.444989\pi$
$619$	5296.63	0.343925	0.171962	+	0.985104i	$0.444989\pi$
$620$	−27821.7	−1.80217
$621$	0	0
$622$	−42430.8	−2.73524
$623$	6745.77	0.433810
$624$	0	0
$625$	−11622.6	−0.743844
$626$	34031.7	2.17281
$627$	0	0
$628$	7544.00	0.479360
$629$	8029.59	0.508999
$630$	0	0
$631$	−15857.8	−1.00046	−0.500228	−	0.865894i	$-0.666751\pi$
$631$	−15857.8	−1.00046	−0.500228	+	0.865894i	$0.666751\pi$
$632$	31891.4	2.00724
$633$	0	0
$634$	−10831.4	−0.678503
$635$	−2918.59	−0.182395
$636$	0	0
$637$	−762.892	−0.0474519
$638$	−4257.97	−0.264223
$639$	0	0
$640$	−26169.0	−1.61628
$641$	−30365.3	−1.87107	−0.935537	−	0.353228i	$-0.885084\pi$
$641$	−30365.3	−1.87107	−0.935537	+	0.353228i	$0.885084\pi$
$642$	0	0
$643$	3746.71	0.229791	0.114896	−	0.993378i	$-0.463347\pi$
$643$	3746.71	0.229791	0.114896	+	0.993378i	$0.463347\pi$
$644$	−8843.44	−0.541119
$645$	0	0
$646$	−21545.4	−1.31222
$647$	13032.7	0.791915	0.395957	−	0.918269i	$-0.370413\pi$
$647$	13032.7	0.791915	0.395957	+	0.918269i	$0.370413\pi$
$648$	0	0
$649$	−1037.97	−0.0627794
$650$	1947.27	0.117505
$651$	0	0
$652$	3536.08	0.212398
$653$	−3096.91	−0.185592	−0.0927960	−	0.995685i	$-0.529580\pi$
$653$	−3096.91	−0.185592	−0.0927960	+	0.995685i	$0.529580\pi$
$654$	0	0
$655$	11027.8	0.657853
$656$	13169.5	0.783813
$657$	0	0
$658$	137.766	0.00816213
$659$	11759.2	0.695103	0.347551	−	0.937661i	$-0.387013\pi$
$659$	11759.2	0.695103	0.347551	+	0.937661i	$0.387013\pi$
$660$	0	0
$661$	2363.12	0.139054	0.0695270	−	0.997580i	$-0.477851\pi$
$661$	2363.12	0.139054	0.0695270	+	0.997580i	$0.477851\pi$
$662$	−29805.4	−1.74988
$663$	0	0
$664$	14738.6	0.861396
$665$	9446.30	0.550845
$666$	0	0
$667$	−21273.7	−1.23496
$668$	8445.19	0.489153
$669$	0	0
$670$	−4797.82	−0.276650
$671$	1911.47	0.109972
$672$	0	0
$673$	−19358.2	−1.10877	−0.554387	−	0.832259i	$-0.687047\pi$
$673$	−19358.2	−1.10877	−0.554387	+	0.832259i	$0.687047\pi$
$674$	897.267	0.0512781
$675$	0	0
$676$	−28131.2	−1.60055
$677$	7931.13	0.450248	0.225124	−	0.974330i	$-0.427721\pi$
$677$	7931.13	0.450248	0.225124	+	0.974330i	$0.427721\pi$
$678$	0	0
$679$	5088.35	0.287589
$680$	−10059.8	−0.567317
$681$	0	0
$682$	3420.86	0.192070
$683$	7289.89	0.408404	0.204202	−	0.978929i	$-0.434540\pi$
$683$	7289.89	0.408404	0.204202	+	0.978929i	$0.434540\pi$
$684$	0	0
$685$	−2364.03	−0.131861
$686$	1623.09	0.0903353
$687$	0	0
$688$	−12550.0	−0.695444
$689$	11467.4	0.634069
$690$	0	0
$691$	−13338.6	−0.734335	−0.367167	−	0.930155i	$-0.619672\pi$
$691$	−13338.6	−0.734335	−0.367167	+	0.930155i	$0.619672\pi$
$692$	42569.6	2.33852
$693$	0	0
$694$	238.840	0.0130637
$695$	−1877.50	−0.102471
$696$	0	0
$697$	−15755.1	−0.856195
$698$	−6334.84	−0.343520
$699$	0	0
$700$	−2662.80	−0.143778
$701$	−12918.5	−0.696042	−0.348021	−	0.937487i	$-0.613146\pi$
$701$	−12918.5	−0.696042	−0.348021	+	0.937487i	$0.613146\pi$
$702$	0	0
$703$	−32581.8	−1.74800
$704$	2755.36	0.147509
$705$	0	0
$706$	15109.4	0.805453
$707$	1882.10	0.100118
$708$	0	0
$709$	2282.78	0.120919	0.0604596	−	0.998171i	$-0.480743\pi$
$709$	2282.78	0.120919	0.0604596	+	0.998171i	$0.480743\pi$
$710$	−2074.38	−0.109648
$711$	0	0
$712$	29150.1	1.53434
$713$	17091.3	0.897721
$714$	0	0
$715$	573.906	0.0300180
$716$	39215.2	2.04684
$717$	0	0
$718$	31521.2	1.63839
$719$	−29360.8	−1.52291	−0.761456	−	0.648217i	$-0.775515\pi$
$719$	−29360.8	−1.52291	−0.761456	+	0.648217i	$0.775515\pi$
$720$	0	0
$721$	7758.48	0.400750
$722$	54967.9	2.83337
$723$	0	0
$724$	6420.96	0.329603
$725$	−6405.60	−0.328135
$726$	0	0
$727$	−18139.1	−0.925368	−0.462684	−	0.886523i	$-0.653114\pi$
$727$	−18139.1	−0.925368	−0.462684	+	0.886523i	$0.653114\pi$
$728$	−3296.64	−0.167832
$729$	0	0
$730$	−42358.6	−2.14762
$731$	15014.1	0.759665
$732$	0	0
$733$	−5292.54	−0.266691	−0.133345	−	0.991070i	$-0.542572\pi$
$733$	−5292.54	−0.266691	−0.133345	+	0.991070i	$0.542572\pi$
$734$	21733.2	1.09290
$735$	0	0
$736$	9611.17	0.481348
$737$	379.163	0.0189507
$738$	0	0
$739$	−25955.7	−1.29201	−0.646007	−	0.763332i	$-0.723562\pi$
$739$	−25955.7	−1.29201	−0.646007	+	0.763332i	$0.723562\pi$
$740$	−34251.8	−1.70151
$741$	0	0
$742$	−24397.5	−1.20709
$743$	−25972.0	−1.28240	−0.641199	−	0.767375i	$-0.721563\pi$
$743$	−25972.0	−1.28240	−0.641199	+	0.767375i	$0.721563\pi$
$744$	0	0
$745$	10996.4	0.540773
$746$	33041.9	1.62165
$747$	0	0
$748$	1789.97	0.0874968
$749$	591.265	0.0288443
$750$	0	0
$751$	6487.01	0.315199	0.157599	−	0.987503i	$-0.449625\pi$
$751$	6487.01	0.315199	0.157599	+	0.987503i	$0.449625\pi$
$752$	116.454	0.00564711
$753$	0	0
$754$	−17855.3	−0.862401
$755$	34873.3	1.68102
$756$	0	0
$757$	22221.6	1.06692	0.533459	−	0.845826i	$-0.320892\pi$
$757$	22221.6	1.06692	0.533459	+	0.845826i	$0.320892\pi$
$758$	−57070.6	−2.73469
$759$	0	0
$760$	40819.8	1.94827
$761$	19820.0	0.944120	0.472060	−	0.881567i	$-0.343511\pi$
$761$	19820.0	0.944120	0.472060	+	0.881567i	$0.343511\pi$
$762$	0	0
$763$	−1219.19	−0.0578473
$764$	50630.6	2.39758
$765$	0	0
$766$	45780.1	2.15940
$767$	−4352.59	−0.204906
$768$	0	0
$769$	−3307.64	−0.155106	−0.0775531	−	0.996988i	$-0.524711\pi$
$769$	−3307.64	−0.155106	−0.0775531	+	0.996988i	$0.524711\pi$
$770$	−1221.02	−0.0571459
$771$	0	0
$772$	70085.0	3.26738
$773$	15964.0	0.742801	0.371401	−	0.928473i	$-0.378878\pi$
$773$	15964.0	0.742801	0.371401	+	0.928473i	$0.378878\pi$
$774$	0	0
$775$	5146.28	0.238529
$776$	21988.0	1.01717
$777$	0	0
$778$	−53840.9	−2.48109
$779$	63929.8	2.94034
$780$	0	0
$781$	163.935	0.00751096
$782$	13914.1	0.636273
$783$	0	0
$784$	1372.00	0.0625000
$785$	5204.06	0.236612
$786$	0	0
$787$	−36394.0	−1.64842	−0.824210	−	0.566284i	$-0.808381\pi$
$787$	−36394.0	−1.64842	−0.824210	+	0.566284i	$0.808381\pi$
$788$	−49329.1	−2.23004
$789$	0	0
$790$	49532.2	2.23073
$791$	7348.38	0.330314
$792$	0	0
$793$	8015.51	0.358940
$794$	−71939.9	−3.21543
$795$	0	0
$796$	−57749.2	−2.57144
$797$	−13384.0	−0.594836	−0.297418	−	0.954747i	$-0.596126\pi$
$797$	−13384.0	−0.594836	−0.297418	+	0.954747i	$0.596126\pi$
$798$	0	0
$799$	−139.318	−0.00616859
$800$	2893.97	0.127896
$801$	0	0
$802$	−6606.06	−0.290858
$803$	3347.53	0.147113
$804$	0	0
$805$	−6100.45	−0.267096
$806$	14345.0	0.626898
$807$	0	0
$808$	8133.03	0.354108
$809$	13971.0	0.607161	0.303580	−	0.952806i	$-0.401818\pi$
$809$	13971.0	0.607161	0.303580	+	0.952806i	$0.401818\pi$
$810$	0	0
$811$	18471.8	0.799793	0.399897	−	0.916560i	$-0.369046\pi$
$811$	18471.8	0.799793	0.399897	+	0.916560i	$0.369046\pi$
$812$	24416.2	1.05522
$813$	0	0
$814$	4211.48	0.181342
$815$	2439.29	0.104840
$816$	0	0
$817$	−60922.8	−2.60884
$818$	−41043.6	−1.75435
$819$	0	0
$820$	67206.5	2.86214
$821$	−28188.3	−1.19827	−0.599135	−	0.800648i	$-0.704489\pi$
$821$	−28188.3	−1.19827	−0.599135	+	0.800648i	$0.704489\pi$
$822$	0	0
$823$	−8071.14	−0.341849	−0.170925	−	0.985284i	$-0.554675\pi$
$823$	−8071.14	−0.341849	−0.170925	+	0.985284i	$0.554675\pi$
$824$	33526.3	1.41741
$825$	0	0
$826$	9260.38	0.390085
$827$	−41952.8	−1.76402	−0.882009	−	0.471233i	$-0.843809\pi$
$827$	−41952.8	−1.76402	−0.882009	+	0.471233i	$0.843809\pi$
$828$	0	0
$829$	14443.2	0.605108	0.302554	−	0.953132i	$-0.402161\pi$
$829$	14443.2	0.605108	0.302554	+	0.953132i	$0.402161\pi$
$830$	22891.2	0.957307
$831$	0	0
$832$	11554.3	0.481457
$833$	−1641.37	−0.0682716
$834$	0	0
$835$	5825.72	0.241446
$836$	−7263.18	−0.300481
$837$	0	0
$838$	34609.9	1.42670
$839$	−26360.6	−1.08471	−0.542353	−	0.840151i	$-0.682466\pi$
$839$	−26360.6	−1.08471	−0.542353	+	0.840151i	$0.682466\pi$
$840$	0	0
$841$	34346.4	1.40827
$842$	−36414.4	−1.49041
$843$	0	0
$844$	61614.5	2.51287
$845$	−19405.7	−0.790030
$846$	0	0
$847$	−9220.51	−0.374050
$848$	−20623.2	−0.835146
$849$	0	0
$850$	4189.58	0.169061
$851$	21041.4	0.847580
$852$	0	0
$853$	−18939.8	−0.760241	−0.380120	−	0.924937i	$-0.624117\pi$
$853$	−18939.8	−0.760241	−0.380120	+	0.924937i	$0.624117\pi$
$854$	−17053.4	−0.683321
$855$	0	0
$856$	2555.00	0.102019
$857$	−44583.6	−1.77707	−0.888533	−	0.458812i	$-0.848275\pi$
$857$	−44583.6	−1.77707	−0.888533	+	0.458812i	$0.848275\pi$
$858$	0	0
$859$	7975.35	0.316782	0.158391	−	0.987376i	$-0.449369\pi$
$859$	7975.35	0.316782	0.158391	+	0.987376i	$0.449369\pi$
$860$	−64045.4	−2.53945
$861$	0	0
$862$	29613.7	1.17012
$863$	8258.42	0.325747	0.162874	−	0.986647i	$-0.447924\pi$
$863$	8258.42	0.325747	0.162874	+	0.986647i	$0.447924\pi$
$864$	0	0
$865$	29365.7	1.15429
$866$	−56421.7	−2.21396
$867$	0	0
$868$	−19616.0	−0.767064
$869$	−3914.45	−0.152806
$870$	0	0
$871$	1589.97	0.0618533
$872$	−5268.40	−0.204599
$873$	0	0
$874$	−56459.3	−2.18508
$875$	−10524.0	−0.406603
$876$	0	0
$877$	−43106.2	−1.65974	−0.829871	−	0.557956i	$-0.811586\pi$
$877$	−43106.2	−1.65974	−0.829871	+	0.557956i	$0.811586\pi$
$878$	80992.1	3.11316
$879$	0	0
$880$	−1032.12	−0.0395374
$881$	662.616	0.0253395	0.0126697	−	0.999920i	$-0.495967\pi$
$881$	662.616	0.0253395	0.0126697	+	0.999920i	$0.495967\pi$
$882$	0	0
$883$	32497.0	1.23852	0.619258	−	0.785187i	$-0.287433\pi$
$883$	32497.0	1.23852	0.619258	+	0.785187i	$0.287433\pi$
$884$	7506.00	0.285582
$885$	0	0
$886$	−38306.1	−1.45251
$887$	3176.34	0.120238	0.0601190	−	0.998191i	$-0.480852\pi$
$887$	3176.34	0.120238	0.0601190	+	0.998191i	$0.480852\pi$
$888$	0	0
$889$	−2057.79	−0.0776332
$890$	45274.5	1.70517
$891$	0	0
$892$	−10997.9	−0.412823
$893$	565.312	0.0211841
$894$	0	0
$895$	27051.7	1.01032
$896$	−18450.8	−0.687943
$897$	0	0
$898$	−29133.8	−1.08264
$899$	−47188.1	−1.75063
$900$	0	0
$901$	24672.3	0.912268
$902$	−8263.48	−0.305037
$903$	0	0
$904$	31754.2	1.16828
$905$	4429.35	0.162692
$906$	0	0
$907$	−31726.2	−1.16147	−0.580733	−	0.814094i	$-0.697234\pi$
$907$	−31726.2	−1.16147	−0.580733	+	0.814094i	$0.697234\pi$
$908$	55732.0	2.03693
$909$	0	0
$910$	−5120.18	−0.186519
$911$	−10882.3	−0.395772	−0.197886	−	0.980225i	$-0.563408\pi$
$911$	−10882.3	−0.395772	−0.197886	+	0.980225i	$0.563408\pi$
$912$	0	0
$913$	−1809.05	−0.0655760
$914$	26770.3	0.968799
$915$	0	0
$916$	−55642.2	−2.00706
$917$	7775.31	0.280004
$918$	0	0
$919$	−29610.9	−1.06286	−0.531432	−	0.847101i	$-0.678346\pi$
$919$	−29610.9	−1.06286	−0.531432	+	0.847101i	$0.678346\pi$
$920$	−26361.5	−0.944689
$921$	0	0
$922$	−6694.02	−0.239106
$923$	687.442	0.0245151
$924$	0	0
$925$	6335.66	0.225206
$926$	−25576.4	−0.907658
$927$	0	0
$928$	−26535.9	−0.938667
$929$	52496.6	1.85399	0.926995	−	0.375074i	$-0.122383\pi$
$929$	52496.6	1.85399	0.926995	+	0.375074i	$0.122383\pi$
$930$	0	0
$931$	6660.23	0.234458
$932$	−77405.9	−2.72051
$933$	0	0
$934$	80221.5	2.81041
$935$	1234.77	0.0431885
$936$	0	0
$937$	−1661.46	−0.0579269	−0.0289634	−	0.999580i	$-0.509221\pi$
$937$	−1661.46	−0.0579269	−0.0289634	+	0.999580i	$0.509221\pi$
$938$	−3382.76	−0.117752
$939$	0	0
$940$	594.287	0.0206207
$941$	43066.9	1.49197	0.745983	−	0.665965i	$-0.231980\pi$
$941$	43066.9	1.49197	0.745983	+	0.665965i	$0.231980\pi$
$942$	0	0
$943$	−41286.0	−1.42572
$944$	7827.79	0.269887
$945$	0	0
$946$	7874.79	0.270647
$947$	−10024.6	−0.343986	−0.171993	−	0.985098i	$-0.555021\pi$
$947$	−10024.6	−0.343986	−0.171993	+	0.985098i	$0.555021\pi$
$948$	0	0
$949$	14037.4	0.480163
$950$	−17000.1	−0.580587
$951$	0	0
$952$	−7092.78	−0.241469
$953$	−36134.8	−1.22825	−0.614125	−	0.789209i	$-0.710491\pi$
$953$	−36134.8	−1.22825	−0.614125	+	0.789209i	$0.710491\pi$
$954$	0	0
$955$	34926.4	1.18345
$956$	68552.7	2.31920
$957$	0	0
$958$	−86940.7	−2.93208
$959$	−1666.79	−0.0561245
$960$	0	0
$961$	8120.07	0.272568
$962$	17660.3	0.591883
$963$	0	0
$964$	−4577.19	−0.152927
$965$	48346.5	1.61278
$966$	0	0
$967$	22016.7	0.732171	0.366086	−	0.930581i	$-0.380698\pi$
$967$	22016.7	0.732171	0.366086	+	0.930581i	$0.380698\pi$
$968$	−39844.1	−1.32297
$969$	0	0
$970$	34150.7	1.13042
$971$	8087.76	0.267300	0.133650	−	0.991029i	$-0.457330\pi$
$971$	8087.76	0.267300	0.133650	+	0.991029i	$0.457330\pi$
$972$	0	0
$973$	−1323.75	−0.0436152
$974$	−201.377	−0.00662478
$975$	0	0
$976$	−14415.3	−0.472768
$977$	−19050.6	−0.623830	−0.311915	−	0.950110i	$-0.600970\pi$
$977$	−19050.6	−0.623830	−0.311915	+	0.950110i	$0.600970\pi$
$978$	0	0
$979$	−3577.97	−0.116805
$980$	7001.60	0.228222
$981$	0	0
$982$	−28828.1	−0.936803
$983$	−35176.7	−1.14136	−0.570682	−	0.821171i	$-0.693321\pi$
$983$	−35176.7	−1.14136	−0.570682	+	0.821171i	$0.693321\pi$
$984$	0	0
$985$	−34028.5	−1.10075
$986$	−38415.9	−1.24078
$987$	0	0
$988$	−30457.2	−0.980742
$989$	39344.1	1.26498
$990$	0	0
$991$	−26830.6	−0.860041	−0.430021	−	0.902819i	$-0.641494\pi$
$991$	−26830.6	−0.860041	−0.430021	+	0.902819i	$0.641494\pi$
$992$	21319.0	0.682337
$993$	0	0
$994$	−1462.57	−0.0466699
$995$	−39837.0	−1.26926
$996$	0	0
$997$	27316.3	0.867721	0.433860	−	0.900980i	$-0.357151\pi$
$997$	27316.3	0.867721	0.433860	+	0.900980i	$0.357151\pi$
$998$	29001.1	0.919852
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.4.a.i.1.2	yes	2
3.2	odd	2		189.4.a.e.1.1	✓	2
7.6	odd	2		1323.4.a.x.1.2		2
21.20	even	2		1323.4.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.a.e.1.1	✓	2	3.2	odd	2
189.4.a.i.1.2	yes	2	1.1	even	1	trivial
1323.4.a.o.1.1		2	21.20	even	2
1323.4.a.x.1.2		2	7.6	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:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	189.a (trivial)

\(f(q)\)	\(=\)	\(q+4.73205 q^{2} +14.3923 q^{4} +9.92820 q^{5} +7.00000 q^{7} +30.2487 q^{8} +O(q^{10})\)	\(q+4.73205 q^{2} +14.3923 q^{4} +9.92820 q^{5} +7.00000 q^{7} +30.2487 q^{8} +46.9808 q^{10} -3.71281 q^{11} -15.5692 q^{13} +33.1244 q^{14} +28.0000 q^{16} -33.4974 q^{17} +135.923 q^{19} +142.890 q^{20} -17.5692 q^{22} -87.7795 q^{23} -26.4308 q^{25} -73.6743 q^{26} +100.746 q^{28} +242.354 q^{29} -194.708 q^{31} -109.492 q^{32} -158.512 q^{34} +69.4974 q^{35} -239.708 q^{37} +643.195 q^{38} +300.315 q^{40} +470.338 q^{41} -448.215 q^{43} -53.4359 q^{44} -415.377 q^{46} +4.15906 q^{47} +49.0000 q^{49} -125.072 q^{50} -224.077 q^{52} -736.543 q^{53} -36.8616 q^{55} +211.741 q^{56} +1146.83 q^{58} +279.564 q^{59} -514.831 q^{61} -921.367 q^{62} -742.123 q^{64} -154.574 q^{65} -102.123 q^{67} -482.105 q^{68} +328.865 q^{70} -44.1539 q^{71} -901.615 q^{73} -1134.31 q^{74} +1956.25 q^{76} -25.9897 q^{77} +1054.31 q^{79} +277.990 q^{80} +2225.67 q^{82} +487.246 q^{83} -332.569 q^{85} -2120.98 q^{86} -112.308 q^{88} +963.682 q^{89} -108.985 q^{91} -1263.35 q^{92} +19.6809 q^{94} +1349.47 q^{95} +726.908 q^{97} +231.870 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{2} + 8 q^{4} + 6 q^{5} + 14 q^{7} + 12 q^{8}+O(q^{10}) \) 2 * q + 6 * q^2 + 8 * q^4 + 6 * q^5 + 14 * q^7 + 12 * q^8	\( 2 q + 6 q^{2} + 8 q^{4} + 6 q^{5} + 14 q^{7} + 12 q^{8} + 42 q^{10} + 48 q^{11} + 52 q^{13} + 42 q^{14} + 56 q^{16} + 30 q^{17} + 64 q^{19} + 168 q^{20} + 48 q^{22} + 60 q^{23} - 136 q^{25} + 12 q^{26} + 56 q^{28} + 360 q^{29} - 140 q^{31} + 72 q^{32} - 78 q^{34} + 42 q^{35} - 230 q^{37} + 552 q^{38} + 372 q^{40} + 234 q^{41} - 938 q^{43} - 384 q^{44} - 228 q^{46} + 618 q^{47} + 98 q^{49} - 264 q^{50} - 656 q^{52} - 420 q^{53} - 240 q^{55} + 84 q^{56} + 1296 q^{58} + 282 q^{59} - 32 q^{61} - 852 q^{62} - 736 q^{64} - 420 q^{65} + 544 q^{67} - 888 q^{68} + 294 q^{70} - 504 q^{71} - 764 q^{73} - 1122 q^{74} + 2416 q^{76} + 336 q^{77} + 238 q^{79} + 168 q^{80} + 1926 q^{82} - 522 q^{83} - 582 q^{85} - 2742 q^{86} - 1056 q^{88} + 708 q^{89} + 364 q^{91} - 2208 q^{92} + 798 q^{94} + 1632 q^{95} + 664 q^{97} + 294 q^{98}+O(q^{100}) \) 2 * q + 6 * q^2 + 8 * q^4 + 6 * q^5 + 14 * q^7 + 12 * q^8 + 42 * q^10 + 48 * q^11 + 52 * q^13 + 42 * q^14 + 56 * q^16 + 30 * q^17 + 64 * q^19 + 168 * q^20 + 48 * q^22 + 60 * q^23 - 136 * q^25 + 12 * q^26 + 56 * q^28 + 360 * q^29 - 140 * q^31 + 72 * q^32 - 78 * q^34 + 42 * q^35 - 230 * q^37 + 552 * q^38 + 372 * q^40 + 234 * q^41 - 938 * q^43 - 384 * q^44 - 228 * q^46 + 618 * q^47 + 98 * q^49 - 264 * q^50 - 656 * q^52 - 420 * q^53 - 240 * q^55 + 84 * q^56 + 1296 * q^58 + 282 * q^59 - 32 * q^61 - 852 * q^62 - 736 * q^64 - 420 * q^65 + 544 * q^67 - 888 * q^68 + 294 * q^70 - 504 * q^71 - 764 * q^73 - 1122 * q^74 + 2416 * q^76 + 336 * q^77 + 238 * q^79 + 168 * q^80 + 1926 * q^82 - 522 * q^83 - 582 * q^85 - 2742 * q^86 - 1056 * q^88 + 708 * q^89 + 364 * q^91 - 2208 * q^92 + 798 * q^94 + 1632 * q^95 + 664 * q^97 + 294 * q^98

Introduction

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