LMFDB - Embedded newform 1323.4.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, 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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1323.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:	$78.0595269376$
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:	no (minimal twist has level 189)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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	$-0.815411\pi$
$2$	−4.73205	−1.67303	−0.836516	+	0.547942i	$0.815411\pi$
$3$	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
$7$	0	0
$7$	0	0
$8$	−30.2487	−1.33682
$9$	0	0
$10$	−46.9808	−1.48566
$11$	3.71281	0.101769	0.0508843	−	0.998705i	$-0.483796\pi$
$11$	3.71281	0.101769	0.0508843	+	0.998705i	$0.483796\pi$
$12$	0	0
$13$	15.5692	0.332163	0.166082	−	0.986112i	$-0.446888\pi$
$13$	15.5692	0.332163	0.166082	+	0.986112i	$0.446888\pi$
$14$	0	0
$15$	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	$-0.806362\pi$
$19$	−135.923	−1.64120	−0.820602	+	0.571500i	$0.806362\pi$
$20$	142.890	1.59756
$21$	0	0
$22$	−17.5692	−0.170262
$23$	87.7795	0.795795	0.397897	−	0.917430i	$-0.369740\pi$
$23$	87.7795	0.795795	0.397897	+	0.917430i	$0.369740\pi$
$24$	0	0
$25$	−26.4308	−0.211446
$26$	−73.6743	−0.555720
$27$	0	0
$28$	0	0
$29$	−242.354	−1.55186	−0.775931	−	0.630818i	$-0.782719\pi$
$29$	−242.354	−1.55186	−0.775931	+	0.630818i	$0.782719\pi$
$30$	0	0
$31$	194.708	1.12808	0.564041	−	0.825747i	$-0.309246\pi$
$31$	194.708	1.12808	0.564041	+	0.825747i	$0.309246\pi$
$32$	109.492	0.604865
$33$	0	0
$34$	158.512	0.799544
$35$	0	0
$36$	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
$49$	0	0
$50$	125.072	0.353756
$51$	0	0
$52$	224.077	0.597575
$53$	736.543	1.90891	0.954453	−	0.298361i	$-0.0964400\pi$
$53$	736.543	1.90891	0.954453	+	0.298361i	$0.0964400\pi$
$54$	0	0
$55$	36.8616	0.0903711
$56$	0	0
$57$	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	$-0.318309\pi$
$61$	514.831	1.08061	0.540306	+	0.841469i	$0.318309\pi$
$62$	−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
$70$	0	0
$71$	44.1539	0.0738043	0.0369021	−	0.999319i	$-0.488251\pi$
$71$	44.1539	0.0738043	0.0369021	+	0.999319i	$0.488251\pi$
$72$	0	0
$73$	901.615	1.44556	0.722781	−	0.691077i	$-0.242863\pi$
$73$	901.615	1.44556	0.722781	+	0.691077i	$0.242863\pi$
$74$	1134.31	1.78190
$75$	0	0
$76$	−1956.25	−2.95259
$77$	0	0
$78$	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
$91$	0	0
$92$	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	$-0.624229\pi$
$97$	−726.908	−0.760890	−0.380445	+	0.924804i	$0.624229\pi$
$98$	0	0
$99$	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	$-0.677862\pi$
$103$	−1108.35	−1.06028	−0.530142	+	0.847909i	$0.677862\pi$
$104$	−470.949	−0.444042
$105$	0	0
$106$	−3485.36	−3.19366
$107$	−84.4665	−0.0763148	−0.0381574	−	0.999272i	$-0.512149\pi$
$107$	−84.4665	−0.0763148	−0.0381574	+	0.999272i	$0.512149\pi$
$108$	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
$112$	0	0
$113$	−1049.77	−0.873929	−0.436964	−	0.899479i	$-0.643946\pi$
$113$	−1049.77	−0.873929	−0.436964	+	0.899479i	$0.643946\pi$
$114$	0	0
$115$	871.492	0.706670
$116$	−3488.03	−2.79186
$117$	0	0
$118$	−1322.91	−1.03207
$119$	0	0
$120$	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
$133$	0	0
$134$	483.251	0.311541
$135$	0	0
$136$	1013.25	0.638866
$137$	238.113	0.148492	0.0742458	−	0.997240i	$-0.476345\pi$
$137$	238.113	0.148492	0.0742458	+	0.997240i	$0.476345\pi$
$138$	0	0
$139$	189.108	0.115395	0.0576974	−	0.998334i	$-0.481624\pi$
$139$	189.108	0.115395	0.0576974	+	0.998334i	$0.481624\pi$
$140$	0	0
$141$	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	$-0.598485\pi$
$149$	−1107.59	−0.608975	−0.304487	+	0.952516i	$0.598485\pi$
$150$	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
$154$	0	0
$155$	1933.10	1.00174
$156$	0	0
$157$	−524.169	−0.266454	−0.133227	−	0.991086i	$-0.542534\pi$
$157$	−524.169	−0.266454	−0.133227	+	0.991086i	$0.542534\pi$
$158$	−4989.04	−2.51207
$159$	0	0
$160$	1087.06	0.537123
$161$	0	0
$162$	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
$175$	0	0
$176$	103.959	0.0445238
$177$	0	0
$178$	−4560.19	−1.92023
$179$	−2724.73	−1.13774	−0.568872	−	0.822426i	$-0.692620\pi$
$179$	−2724.73	−1.13774	−0.568872	+	0.822426i	$0.692620\pi$
$180$	0	0
$181$	−446.138	−0.183211	−0.0916055	−	0.995795i	$-0.529200\pi$
$181$	−446.138	−0.183211	−0.0916055	+	0.995795i	$0.529200\pi$
$182$	0	0
$183$	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	$-0.732145\pi$
$191$	−3517.90	−1.33270	−0.666351	+	0.745638i	$0.732145\pi$
$192$	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
$196$	0	0
$197$	3427.46	1.23958	0.619788	−	0.784769i	$-0.287219\pi$
$197$	3427.46	1.23958	0.619788	+	0.784769i	$0.287219\pi$
$198$	0	0
$199$	4012.51	1.42934	0.714671	−	0.699461i	$-0.246576\pi$
$199$	4012.51	1.42934	0.714671	+	0.699461i	$0.246576\pi$
$200$	799.497	0.282665
$201$	0	0
$202$	−1272.32	−0.443167
$203$	0	0
$204$	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
$217$	0	0
$218$	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	$-0.463398\pi$
$223$	764.153	0.229469	0.114734	+	0.993396i	$0.463398\pi$
$224$	0	0
$225$	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	$-0.311640\pi$
$229$	3866.11	1.11563	0.557816	+	0.829965i	$0.311640\pi$
$230$	−4123.95	−1.18228
$231$	0	0
$232$	7330.89	2.07455
$233$	5378.29	1.51220	0.756102	−	0.654454i	$-0.227102\pi$
$233$	5378.29	1.51220	0.756102	+	0.654454i	$0.227102\pi$
$234$	0	0
$235$	41.2920	0.0114621
$236$	4023.57	1.10980
$237$	0	0
$238$	0	0
$239$	−4763.15	−1.28913	−0.644566	−	0.764548i	$-0.722962\pi$
$239$	−4763.15	−1.28913	−0.644566	+	0.764548i	$0.722962\pi$
$240$	0	0
$241$	318.030	0.0850047	0.0425023	−	0.999096i	$-0.486467\pi$
$241$	318.030	0.0850047	0.0425023	+	0.999096i	$0.486467\pi$
$242$	6233.13	1.65571
$243$	0	0
$244$	7409.60	1.94406
$245$	0	0
$246$	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
$259$	0	0
$260$	2224.68	0.530650
$261$	0	0
$262$	−5256.17	−1.23942
$263$	3654.73	0.856883	0.428441	−	0.903570i	$-0.359063\pi$
$263$	3654.73	0.856883	0.428441	+	0.903570i	$0.359063\pi$
$264$	0	0
$265$	7312.55	1.69512
$266$	0	0
$267$	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	$-0.544056\pi$
$271$	−1230.98	−0.275930	−0.137965	+	0.990437i	$0.544056\pi$
$272$	−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
$280$	0	0
$281$	3615.71	0.767598	0.383799	−	0.923417i	$-0.374616\pi$
$281$	3615.71	0.767598	0.383799	+	0.923417i	$0.374616\pi$
$282$	0	0
$283$	2353.42	0.494332	0.247166	−	0.968973i	$-0.420501\pi$
$283$	2353.42	0.494332	0.247166	+	0.968973i	$0.420501\pi$
$284$	635.476	0.132777
$285$	0	0
$286$	−273.539	−0.0565549
$287$	0	0
$288$	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
$301$	0	0
$302$	−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	$-0.180303\pi$
$307$	9077.92	1.68764	0.843818	+	0.536629i	$0.180303\pi$
$308$	0	0
$309$	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	$-0.724964\pi$
$313$	−7191.74	−1.29873	−0.649363	+	0.760479i	$0.724964\pi$
$314$	2480.39	0.445786
$315$	0	0
$316$	15173.9	2.70126
$317$	2288.95	0.405553	0.202776	−	0.979225i	$-0.435004\pi$
$317$	2288.95	0.405553	0.202776	+	0.979225i	$0.435004\pi$
$318$	0	0
$319$	−899.814	−0.157931
$320$	−7367.95	−1.28713
$321$	0	0
$322$	0	0
$323$	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
$329$	0	0
$330$	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
$343$	0	0
$344$	13557.9	2.12499
$345$	0	0
$346$	−13996.5	−2.17473
$347$	−50.4728	−0.00780841	−0.00390421	−	0.999992i	$-0.501243\pi$
$347$	−50.4728	−0.00780841	−0.00390421	+	0.999992i	$0.501243\pi$
$348$	0	0
$349$	1338.71	0.205328	0.102664	−	0.994716i	$-0.467263\pi$
$349$	1338.71	0.205328	0.102664	+	0.994716i	$0.467263\pi$
$350$	0	0
$351$	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	$-0.662874\pi$
$359$	−6661.22	−0.979292	−0.489646	+	0.871921i	$0.662874\pi$
$360$	0	0
$361$	11616.1	1.69355
$362$	2111.15	0.306518
$363$	0	0
$364$	0	0
$365$	8951.42	1.28367
$366$	0	0
$367$	−4592.77	−0.653244	−0.326622	−	0.945155i	$-0.605910\pi$
$367$	−4592.77	−0.653244	−0.326622	+	0.945155i	$0.605910\pi$
$368$	2457.82	0.348160
$369$	0	0
$370$	11261.6	1.58234
$371$	0	0
$372$	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
$385$	0	0
$386$	−23043.3	−3.03853
$387$	0	0
$388$	−10461.9	−1.36887
$389$	11377.9	1.48299	0.741496	−	0.670957i	$-0.234117\pi$
$389$	11377.9	1.48299	0.741496	+	0.670957i	$0.234117\pi$
$390$	0	0
$391$	−2940.39	−0.380311
$392$	0	0
$393$	0	0
$394$	−16218.9	−2.07385
$395$	10467.4	1.33334
$396$	0	0
$397$	15202.7	1.92192	0.960958	−	0.276693i	$-0.0892384\pi$
$397$	15202.7	1.92192	0.960958	+	0.276693i	$0.0892384\pi$
$398$	−18987.4	−2.39134
$399$	0	0
$400$	−740.062	−0.0925077
$401$	1396.02	0.173851	0.0869254	−	0.996215i	$-0.472296\pi$
$401$	1396.02	0.173851	0.0869254	+	0.996215i	$0.472296\pi$
$402$	0	0
$403$	3031.45	0.374707
$404$	3869.69	0.476545
$405$	0	0
$406$	0	0
$407$	−889.990	−0.108391
$408$	0	0
$409$	8673.54	1.04860	0.524302	−	0.851533i	$-0.324326\pi$
$409$	8673.54	1.04860	0.524302	+	0.851533i	$0.324326\pi$
$410$	−22096.9	−2.66167
$411$	0	0
$412$	−15951.8	−1.90749
$413$	0	0
$414$	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
$427$	0	0
$428$	−1215.67	−0.137293
$429$	0	0
$430$	21057.5	2.36159
$431$	−6258.10	−0.699402	−0.349701	−	0.936861i	$-0.613717\pi$
$431$	−6258.10	−0.699402	−0.349701	+	0.936861i	$0.613717\pi$
$432$	0	0
$433$	11923.3	1.32332	0.661660	−	0.749804i	$-0.269852\pi$
$433$	11923.3	1.32332	0.661660	+	0.749804i	$0.269852\pi$
$434$	0	0
$435$	0	0
$436$	−2506.70	−0.275342
$437$	−11931.3	−1.30606
$438$	0	0
$439$	−17115.6	−1.86079	−0.930393	−	0.366564i	$-0.880534\pi$
$439$	−17115.6	−1.86079	−0.930393	+	0.366564i	$0.880534\pi$
$440$	−1115.01	−0.120810
$441$	0	0
$442$	2467.90	0.265579
$443$	8095.04	0.868187	0.434094	−	0.900868i	$-0.357069\pi$
$443$	8095.04	0.868187	0.434094	+	0.900868i	$0.357069\pi$
$444$	0	0
$445$	9567.63	1.01921
$446$	−3616.01	−0.383908
$447$	0	0
$448$	0	0
$449$	6156.71	0.647111	0.323556	−	0.946209i	$-0.395122\pi$
$449$	6156.71	0.647111	0.323556	+	0.946209i	$0.395122\pi$
$450$	0	0
$451$	1746.28	0.182326
$452$	−15108.6	−1.57223
$453$	0	0
$454$	−18324.2	−1.89426
$455$	0	0
$456$	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
$469$	0	0
$470$	−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
$476$	0	0
$477$	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
$490$	0	0
$491$	6092.09	0.559943	0.279972	−	0.960008i	$-0.409675\pi$
$491$	6092.09	0.559943	0.279972	+	0.960008i	$0.409675\pi$
$492$	0	0
$493$	8118.23	0.741636
$494$	10014.0	0.912051
$495$	0	0
$496$	5451.81	0.493536
$497$	0	0
$498$	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
$511$	0	0
$512$	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
$518$	0	0
$519$	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	$-0.718462\pi$
$523$	−15158.7	−1.26739	−0.633693	+	0.773584i	$0.718462\pi$
$524$	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
$532$	0	0
$533$	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
$539$	0	0
$540$	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
$553$	0	0
$554$	−18900.6	−1.44948
$555$	0	0
$556$	2721.69	0.207600
$557$	−13975.1	−1.06309	−0.531547	−	0.847029i	$-0.678389\pi$
$557$	−13975.1	−1.06309	−0.531547	+	0.847029i	$0.678389\pi$
$558$	0	0
$559$	−6978.36	−0.528002
$560$	0	0
$561$	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	$-0.678707\pi$
$569$	−14452.1	−1.06478	−0.532392	+	0.846498i	$0.678707\pi$
$570$	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
$574$	0	0
$575$	−2320.08	−0.168268
$576$	0	0
$577$	−12251.2	−0.883923	−0.441961	−	0.897034i	$-0.645717\pi$
$577$	−12251.2	−0.883923	−0.441961	+	0.897034i	$0.645717\pi$
$578$	17938.8	1.29093
$579$	0	0
$580$	−34629.9	−2.47918
$581$	0	0
$582$	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
$595$	0	0
$596$	−15940.8	−1.09557
$597$	0	0
$598$	−6467.09	−0.442239
$599$	16830.3	1.14803	0.574013	−	0.818846i	$-0.305386\pi$
$599$	16830.3	1.14803	0.574013	+	0.818846i	$0.305386\pi$
$600$	0	0
$601$	−8783.94	−0.596180	−0.298090	−	0.954538i	$-0.596350\pi$
$601$	−8783.94	−0.596180	−0.298090	+	0.954538i	$0.596350\pi$
$602$	0	0
$603$	0	0
$604$	50553.7	3.40563
$605$	−13077.6	−0.878809
$606$	0	0
$607$	−16803.3	−1.12360	−0.561799	−	0.827274i	$-0.689891\pi$
$607$	−16803.3	−1.12360	−0.561799	+	0.827274i	$0.689891\pi$
$608$	−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
$616$	0	0
$617$	2586.39	0.168759	0.0843793	−	0.996434i	$-0.473109\pi$
$617$	2586.39	0.168759	0.0843793	+	0.996434i	$0.473109\pi$
$618$	0	0
$619$	−5296.63	−0.343925	−0.171962	−	0.985104i	$-0.555011\pi$
$619$	−5296.63	−0.343925	−0.171962	+	0.985104i	$0.555011\pi$
$620$	27821.7	1.80217
$621$	0	0
$622$	42430.8	2.73524
$623$	0	0
$624$	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
$637$	0	0
$638$	4257.97	0.264223
$639$	0	0
$640$	26169.0	1.61628
$641$	30365.3	1.87107	0.935537	−	0.353228i	$-0.114916\pi$
$641$	30365.3	1.87107	0.935537	+	0.353228i	$0.114916\pi$
$642$	0	0
$643$	−3746.71	−0.229791	−0.114896	−	0.993378i	$-0.536653\pi$
$643$	−3746.71	−0.229791	−0.114896	+	0.993378i	$0.536653\pi$
$644$	0	0
$645$	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	$-0.470420\pi$
$653$	3096.91	0.185592	0.0927960	+	0.995685i	$0.470420\pi$
$654$	0	0
$655$	11027.8	0.657853
$656$	13169.5	0.783813
$657$	0	0
$658$	0	0
$659$	−11759.2	−0.695103	−0.347551	−	0.937661i	$-0.612987\pi$
$659$	−11759.2	−0.695103	−0.347551	+	0.937661i	$0.612987\pi$
$660$	0	0
$661$	−2363.12	−0.139054	−0.0695270	−	0.997580i	$-0.522149\pi$
$661$	−2363.12	−0.139054	−0.0695270	+	0.997580i	$0.522149\pi$
$662$	29805.4	1.74988
$663$	0	0
$664$	−14738.6	−0.861396
$665$	0	0
$666$	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
$679$	0	0
$680$	10059.8	0.567317
$681$	0	0
$682$	−3420.86	−0.192070
$683$	−7289.89	−0.408404	−0.204202	−	0.978929i	$-0.565460\pi$
$683$	−7289.89	−0.408404	−0.204202	+	0.978929i	$0.565460\pi$
$684$	0	0
$685$	2364.03	0.131861
$686$	0	0
$687$	0	0
$688$	−12550.0	−0.695444
$689$	11467.4	0.634069
$690$	0	0
$691$	13338.6	0.734335	0.367167	−	0.930155i	$-0.380328\pi$
$691$	13338.6	0.734335	0.367167	+	0.930155i	$0.380328\pi$
$692$	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
$700$	0	0
$701$	12918.5	0.696042	0.348021	−	0.937487i	$-0.386854\pi$
$701$	12918.5	0.696042	0.348021	+	0.937487i	$0.386854\pi$
$702$	0	0
$703$	32581.8	1.74800
$704$	−2755.36	−0.147509
$705$	0	0
$706$	−15109.4	−0.805453
$707$	0	0
$708$	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
$721$	0	0
$722$	−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	$-0.346886\pi$
$727$	18139.1	0.925368	0.462684	+	0.886523i	$0.346886\pi$
$728$	0	0
$729$	0	0
$730$	−42358.6	−2.14762
$731$	15014.1	0.759665
$732$	0	0
$733$	5292.54	0.266691	0.133345	−	0.991070i	$-0.457428\pi$
$733$	5292.54	0.266691	0.133345	+	0.991070i	$0.457428\pi$
$734$	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
$742$	0	0
$743$	25972.0	1.28240	0.641199	−	0.767375i	$-0.278437\pi$
$743$	25972.0	1.28240	0.641199	+	0.767375i	$0.278437\pi$
$744$	0	0
$745$	−10996.4	−0.540773
$746$	−33041.9	−1.62165
$747$	0	0
$748$	−1789.97	−0.0874968
$749$	0	0
$750$	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
$763$	0	0
$764$	−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	$-0.475289\pi$
$769$	3307.64	0.155106	0.0775531	+	0.996988i	$0.475289\pi$
$770$	0	0
$771$	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
$784$	0	0
$785$	−5204.06	−0.236612
$786$	0	0
$787$	36394.0	1.64842	0.824210	−	0.566284i	$-0.191619\pi$
$787$	36394.0	1.64842	0.824210	+	0.566284i	$0.191619\pi$
$788$	49329.1	2.23004
$789$	0	0
$790$	−49532.2	−2.23073
$791$	0	0
$792$	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
$805$	0	0
$806$	−14345.0	−0.626898
$807$	0	0
$808$	−8133.03	−0.354108
$809$	−13971.0	−0.607161	−0.303580	−	0.952806i	$-0.598182\pi$
$809$	−13971.0	−0.607161	−0.303580	+	0.952806i	$0.598182\pi$
$810$	0	0
$811$	−18471.8	−0.799793	−0.399897	−	0.916560i	$-0.630954\pi$
$811$	−18471.8	−0.799793	−0.399897	+	0.916560i	$0.630954\pi$
$812$	0	0
$813$	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	$-0.295511\pi$
$821$	28188.3	1.19827	0.599135	+	0.800648i	$0.295511\pi$
$822$	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
$826$	0	0
$827$	41952.8	1.76402	0.882009	−	0.471233i	$-0.156191\pi$
$827$	41952.8	1.76402	0.882009	+	0.471233i	$0.156191\pi$
$828$	0	0
$829$	−14443.2	−0.605108	−0.302554	−	0.953132i	$-0.597839\pi$
$829$	−14443.2	−0.605108	−0.302554	+	0.953132i	$0.597839\pi$
$830$	−22891.2	−0.957307
$831$	0	0
$832$	−11554.3	−0.481457
$833$	0	0
$834$	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
$847$	0	0
$848$	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	$-0.375883\pi$
$853$	18939.8	0.760241	0.380120	+	0.924937i	$0.375883\pi$
$854$	0	0
$855$	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	$-0.550631\pi$
$859$	−7975.35	−0.316782	−0.158391	+	0.987376i	$0.550631\pi$
$860$	−64045.4	−2.53945
$861$	0	0
$862$	29613.7	1.17012
$863$	−8258.42	−0.325747	−0.162874	−	0.986647i	$-0.552076\pi$
$863$	−8258.42	−0.325747	−0.162874	+	0.986647i	$0.552076\pi$
$864$	0	0
$865$	29365.7	1.15429
$866$	−56421.7	−2.21396
$867$	0	0
$868$	0	0
$869$	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
$875$	0	0
$876$	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
$889$	0	0
$890$	−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
$896$	0	0
$897$	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
$910$	0	0
$911$	10882.3	0.395772	0.197886	−	0.980225i	$-0.436592\pi$
$911$	10882.3	0.395772	0.197886	+	0.980225i	$0.436592\pi$
$912$	0	0
$913$	1809.05	0.0655760
$914$	−26770.3	−0.968799
$915$	0	0
$916$	55642.2	2.00706
$917$	0	0
$918$	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
$931$	0	0
$932$	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	$-0.490779\pi$
$937$	1661.46	0.0579269	0.0289634	+	0.999580i	$0.490779\pi$
$938$	0	0
$939$	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	$-0.444979\pi$
$947$	10024.6	0.343986	0.171993	+	0.985098i	$0.444979\pi$
$948$	0	0
$949$	14037.4	0.480163
$950$	−17000.1	−0.580587
$951$	0	0
$952$	0	0
$953$	36134.8	1.22825	0.614125	−	0.789209i	$-0.289509\pi$
$953$	36134.8	1.22825	0.614125	+	0.789209i	$0.289509\pi$
$954$	0	0
$955$	−34926.4	−1.18345
$956$	−68552.7	−2.31920
$957$	0	0
$958$	86940.7	2.93208
$959$	0	0
$960$	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
$973$	0	0
$974$	201.377	0.00662478
$975$	0	0
$976$	14415.3	0.472768
$977$	19050.6	0.623830	0.311915	−	0.950110i	$-0.399030\pi$
$977$	19050.6	0.623830	0.311915	+	0.950110i	$0.399030\pi$
$978$	0	0
$979$	3577.97	0.116805
$980$	0	0
$981$	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
$994$	0	0
$995$	39837.0	1.26926
$996$	0	0
$997$	−27316.3	−0.867721	−0.433860	−	0.900980i	$-0.642849\pi$
$997$	−27316.3	−0.867721	−0.433860	+	0.900980i	$0.642849\pi$
$998$	−29001.1	−0.919852
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.a.e.1.1	✓	2	7.6	odd	2
189.4.a.i.1.2	yes	2	21.20	even	2
1323.4.a.o.1.1		2	1.1	even	1	trivial
1323.4.a.x.1.2		2	3.2	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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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