LMFDB - Embedded newform 5733.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5733,2,Mod(1,5733)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5733, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5733.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5733 = 3^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5733.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:	$45.7782354788$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	5733.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.81361 q^{2} +1.28917 q^{4} +2.81361 q^{5} -1.28917 q^{8} +O(q^{10})$	$q+1.81361 q^{2} +1.28917 q^{4} +2.81361 q^{5} -1.28917 q^{8} +5.10278 q^{10} -3.10278 q^{11} -1.00000 q^{13} -4.91638 q^{16} -0.524438 q^{17} -0.813607 q^{19} +3.62721 q^{20} -5.62721 q^{22} -7.33804 q^{23} +2.91638 q^{25} -1.81361 q^{26} -8.28917 q^{29} -1.39194 q^{31} -6.33804 q^{32} -0.951124 q^{34} -6.15165 q^{37} -1.47556 q^{38} -3.62721 q^{40} -4.20555 q^{41} +6.75971 q^{43} -4.00000 q^{44} -13.3083 q^{46} -5.97028 q^{47} +5.28917 q^{50} -1.28917 q^{52} +2.49472 q^{53} -8.72999 q^{55} -15.0333 q^{58} -4.47054 q^{59} +2.00000 q^{61} -2.52444 q^{62} -1.66196 q^{64} -2.81361 q^{65} +10.0383 q^{67} -0.676089 q^{68} +8.72999 q^{71} +2.34307 q^{73} -11.1567 q^{74} -1.04888 q^{76} -13.5436 q^{79} -13.8328 q^{80} -7.62721 q^{82} +16.4791 q^{83} -1.47556 q^{85} +12.2594 q^{86} +4.00000 q^{88} -10.6464 q^{89} -9.45998 q^{92} -10.8277 q^{94} -2.28917 q^{95} +1.18639 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + 3 * q^4 + 2 * q^5 - 3 * q^8	$ 3 q - q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{8} + 8 q^{10} - 2 q^{11} - 3 q^{13} - q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} - 4 q^{22} - 10 q^{23} - 5 q^{25} + q^{26} - 24 q^{29} + 4 q^{31} - 7 q^{32} - 14 q^{34} - 10 q^{38} + 2 q^{40} + 2 q^{41} + 10 q^{43} - 12 q^{44} - 18 q^{46} - 8 q^{47} + 15 q^{50} - 3 q^{52} - 8 q^{53} - 6 q^{55} + 12 q^{58} - 4 q^{59} + 6 q^{61} - 2 q^{62} - 17 q^{64} - 2 q^{65} - 12 q^{67} + 22 q^{68} + 6 q^{71} + 10 q^{73} - 30 q^{74} + 8 q^{76} - 14 q^{79} - 14 q^{80} - 10 q^{82} - 12 q^{83} - 10 q^{85} + 26 q^{86} + 12 q^{88} + 2 q^{89} + 12 q^{92} + 10 q^{94} - 6 q^{95} + 10 q^{97}+O(q^{100}) $ 3 * q - q^2 + 3 * q^4 + 2 * q^5 - 3 * q^8 + 8 * q^10 - 2 * q^11 - 3 * q^13 - q^16 + 4 * q^17 + 4 * q^19 - 2 * q^20 - 4 * q^22 - 10 * q^23 - 5 * q^25 + q^26 - 24 * q^29 + 4 * q^31 - 7 * q^32 - 14 * q^34 - 10 * q^38 + 2 * q^40 + 2 * q^41 + 10 * q^43 - 12 * q^44 - 18 * q^46 - 8 * q^47 + 15 * q^50 - 3 * q^52 - 8 * q^53 - 6 * q^55 + 12 * q^58 - 4 * q^59 + 6 * q^61 - 2 * q^62 - 17 * q^64 - 2 * q^65 - 12 * q^67 + 22 * q^68 + 6 * q^71 + 10 * q^73 - 30 * q^74 + 8 * q^76 - 14 * q^79 - 14 * q^80 - 10 * q^82 - 12 * q^83 - 10 * q^85 + 26 * q^86 + 12 * q^88 + 2 * q^89 + 12 * q^92 + 10 * q^94 - 6 * q^95 + 10 * q^97

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 $

$2$	1.81361	1.28241	0.641207	−	0.767368i	$-0.278434\pi$
$2$	1.81361	1.28241	0.641207	+	0.767368i	$0.278434\pi$
$3$	0	0
$3$	0	0
$4$	1.28917	0.644584
$5$	2.81361	1.25828	0.629142	−	0.777291i	$-0.283407\pi$
$5$	2.81361	1.25828	0.629142	+	0.777291i	$0.283407\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.28917	−0.455790
$9$	0	0
$10$	5.10278	1.61364
$11$	−3.10278	−0.935522	−0.467761	−	0.883855i	$-0.654939\pi$
$11$	−3.10278	−0.935522	−0.467761	+	0.883855i	$0.654939\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	−4.91638	−1.22910
$17$	−0.524438	−0.127195	−0.0635974	−	0.997976i	$-0.520257\pi$
$17$	−0.524438	−0.127195	−0.0635974	+	0.997976i	$0.520257\pi$
$18$	0	0
$19$	−0.813607	−0.186654	−0.0933271	−	0.995636i	$-0.529750\pi$
$19$	−0.813607	−0.186654	−0.0933271	+	0.995636i	$0.529750\pi$
$20$	3.62721	0.811069
$21$	0	0
$22$	−5.62721	−1.19973
$23$	−7.33804	−1.53009	−0.765044	−	0.643978i	$-0.777283\pi$
$23$	−7.33804	−1.53009	−0.765044	+	0.643978i	$0.777283\pi$
$24$	0	0
$25$	2.91638	0.583276
$26$	−1.81361	−0.355677
$27$	0	0
$28$	0	0
$29$	−8.28917	−1.53926	−0.769630	−	0.638490i	$-0.779559\pi$
$29$	−8.28917	−1.53926	−0.769630	+	0.638490i	$0.779559\pi$
$30$	0	0
$31$	−1.39194	−0.250000	−0.125000	−	0.992157i	$-0.539893\pi$
$31$	−1.39194	−0.250000	−0.125000	+	0.992157i	$0.539893\pi$
$32$	−6.33804	−1.12042
$33$	0	0
$34$	−0.951124	−0.163116
$35$	0	0
$36$	0	0
$37$	−6.15165	−1.01133	−0.505663	−	0.862731i	$-0.668752\pi$
$37$	−6.15165	−1.01133	−0.505663	+	0.862731i	$0.668752\pi$
$38$	−1.47556	−0.239368
$39$	0	0
$40$	−3.62721	−0.573513
$41$	−4.20555	−0.656797	−0.328398	−	0.944539i	$-0.606509\pi$
$41$	−4.20555	−0.656797	−0.328398	+	0.944539i	$0.606509\pi$
$42$	0	0
$43$	6.75971	1.03085	0.515423	−	0.856936i	$-0.327635\pi$
$43$	6.75971	1.03085	0.515423	+	0.856936i	$0.327635\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−13.3083	−1.96221
$47$	−5.97028	−0.870855	−0.435427	−	0.900224i	$-0.643403\pi$
$47$	−5.97028	−0.870855	−0.435427	+	0.900224i	$0.643403\pi$
$48$	0	0
$49$	0	0
$50$	5.28917	0.748001
$51$	0	0
$52$	−1.28917	−0.178776
$53$	2.49472	0.342676	0.171338	−	0.985212i	$-0.445191\pi$
$53$	2.49472	0.342676	0.171338	+	0.985212i	$0.445191\pi$
$54$	0	0
$55$	−8.72999	−1.17715
$56$	0	0
$57$	0	0
$58$	−15.0333	−1.97397
$59$	−4.47054	−0.582015	−0.291007	−	0.956721i	$-0.593990\pi$
$59$	−4.47054	−0.582015	−0.291007	+	0.956721i	$0.593990\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−2.52444	−0.320604
$63$	0	0
$64$	−1.66196	−0.207744
$65$	−2.81361	−0.348985
$66$	0	0
$67$	10.0383	1.22638	0.613188	−	0.789937i	$-0.289887\pi$
$67$	10.0383	1.22638	0.613188	+	0.789937i	$0.289887\pi$
$68$	−0.676089	−0.0819878
$69$	0	0
$70$	0	0
$71$	8.72999	1.03606	0.518029	−	0.855363i	$-0.326666\pi$
$71$	8.72999	1.03606	0.518029	+	0.855363i	$0.326666\pi$
$72$	0	0
$73$	2.34307	0.274235	0.137118	−	0.990555i	$-0.456216\pi$
$73$	2.34307	0.274235	0.137118	+	0.990555i	$0.456216\pi$
$74$	−11.1567	−1.29694
$75$	0	0
$76$	−1.04888	−0.120314
$77$	0	0
$78$	0	0
$79$	−13.5436	−1.52377	−0.761887	−	0.647710i	$-0.775727\pi$
$79$	−13.5436	−1.52377	−0.761887	+	0.647710i	$0.775727\pi$
$80$	−13.8328	−1.54655
$81$	0	0
$82$	−7.62721	−0.842285
$83$	16.4791	1.80882	0.904410	−	0.426665i	$-0.140312\pi$
$83$	16.4791	1.80882	0.904410	+	0.426665i	$0.140312\pi$
$84$	0	0
$85$	−1.47556	−0.160047
$86$	12.2594	1.32197
$87$	0	0
$88$	4.00000	0.426401
$89$	−10.6464	−1.12851	−0.564256	−	0.825600i	$-0.690837\pi$
$89$	−10.6464	−1.12851	−0.564256	+	0.825600i	$0.690837\pi$
$90$	0	0
$91$	0	0
$92$	−9.45998	−0.986271
$93$	0	0
$94$	−10.8277	−1.11680
$95$	−2.28917	−0.234864
$96$	0	0
$97$	1.18639	0.120460	0.0602300	−	0.998185i	$-0.480817\pi$
$97$	1.18639	0.120460	0.0602300	+	0.998185i	$0.480817\pi$
$98$	0	0
$99$	0	0
$100$	3.75971	0.375971
$101$	13.1028	1.30377	0.651887	−	0.758316i	$-0.273977\pi$
$101$	13.1028	1.30377	0.651887	+	0.758316i	$0.273977\pi$
$102$	0	0
$103$	−4.41110	−0.434639	−0.217319	−	0.976101i	$-0.569731\pi$
$103$	−4.41110	−0.434639	−0.217319	+	0.976101i	$0.569731\pi$
$104$	1.28917	0.126413
$105$	0	0
$106$	4.52444	0.439452
$107$	−0.578337	−0.0559100	−0.0279550	−	0.999609i	$-0.508900\pi$
$107$	−0.578337	−0.0559100	−0.0279550	+	0.999609i	$0.508900\pi$
$108$	0	0
$109$	5.57331	0.533827	0.266913	−	0.963721i	$-0.413996\pi$
$109$	5.57331	0.533827	0.266913	+	0.963721i	$0.413996\pi$
$110$	−15.8328	−1.50959
$111$	0	0
$112$	0	0
$113$	−5.44584	−0.512302	−0.256151	−	0.966637i	$-0.582454\pi$
$113$	−5.44584	−0.512302	−0.256151	+	0.966637i	$0.582454\pi$
$114$	0	0
$115$	−20.6464	−1.92528
$116$	−10.6861	−0.992183
$117$	0	0
$118$	−8.10780	−0.746383
$119$	0	0
$120$	0	0
$121$	−1.37279	−0.124799
$122$	3.62721	0.328392
$123$	0	0
$124$	−1.79445	−0.161146
$125$	−5.86248	−0.524356
$126$	0	0
$127$	−12.8816	−1.14306	−0.571530	−	0.820581i	$-0.693650\pi$
$127$	−12.8816	−1.14306	−0.571530	+	0.820581i	$0.693650\pi$
$128$	9.66196	0.854004
$129$	0	0
$130$	−5.10278	−0.447543
$131$	9.04888	0.790604	0.395302	−	0.918551i	$-0.370640\pi$
$131$	9.04888	0.790604	0.395302	+	0.918551i	$0.370640\pi$
$132$	0	0
$133$	0	0
$134$	18.2056	1.57272
$135$	0	0
$136$	0.676089	0.0579741
$137$	6.25945	0.534781	0.267390	−	0.963588i	$-0.413839\pi$
$137$	6.25945	0.534781	0.267390	+	0.963588i	$0.413839\pi$
$138$	0	0
$139$	11.5733	0.981636	0.490818	−	0.871262i	$-0.336698\pi$
$139$	11.5733	0.981636	0.490818	+	0.871262i	$0.336698\pi$
$140$	0	0
$141$	0	0
$142$	15.8328	1.32866
$143$	3.10278	0.259467
$144$	0	0
$145$	−23.3225	−1.93682
$146$	4.24940	0.351683
$147$	0	0
$148$	−7.93051	−0.651884
$149$	−8.52444	−0.698349	−0.349175	−	0.937058i	$-0.613538\pi$
$149$	−8.52444	−0.698349	−0.349175	+	0.937058i	$0.613538\pi$
$150$	0	0
$151$	−11.9844	−0.975278	−0.487639	−	0.873045i	$-0.662142\pi$
$151$	−11.9844	−0.975278	−0.487639	+	0.873045i	$0.662142\pi$
$152$	1.04888	0.0850751
$153$	0	0
$154$	0	0
$155$	−3.91638	−0.314571
$156$	0	0
$157$	12.8277	1.02377	0.511883	−	0.859055i	$-0.328948\pi$
$157$	12.8277	1.02377	0.511883	+	0.859055i	$0.328948\pi$
$158$	−24.5628	−1.95411
$159$	0	0
$160$	−17.8328	−1.40980
$161$	0	0
$162$	0	0
$163$	−13.4600	−1.05427	−0.527133	−	0.849783i	$-0.676733\pi$
$163$	−13.4600	−1.05427	−0.527133	+	0.849783i	$0.676733\pi$
$164$	−5.42166	−0.423361
$165$	0	0
$166$	29.8867	2.31965
$167$	−2.02972	−0.157064	−0.0785322	−	0.996912i	$-0.525023\pi$
$167$	−2.02972	−0.157064	−0.0785322	+	0.996912i	$0.525023\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−2.67609	−0.205247
$171$	0	0
$172$	8.71440	0.664467
$173$	20.2978	1.54321	0.771605	−	0.636102i	$-0.219454\pi$
$173$	20.2978	1.54321	0.771605	+	0.636102i	$0.219454\pi$
$174$	0	0
$175$	0	0
$176$	15.2544	1.14985
$177$	0	0
$178$	−19.3083	−1.44722
$179$	11.0036	0.822445	0.411223	−	0.911535i	$-0.365102\pi$
$179$	11.0036	0.822445	0.411223	+	0.911535i	$0.365102\pi$
$180$	0	0
$181$	−0.691675	−0.0514118	−0.0257059	−	0.999670i	$-0.508183\pi$
$181$	−0.691675	−0.0514118	−0.0257059	+	0.999670i	$0.508183\pi$
$182$	0	0
$183$	0	0
$184$	9.45998	0.697399
$185$	−17.3083	−1.27253
$186$	0	0
$187$	1.62721	0.118994
$188$	−7.69670	−0.561339
$189$	0	0
$190$	−4.15165	−0.301192
$191$	−7.83276	−0.566759	−0.283379	−	0.959008i	$-0.591456\pi$
$191$	−7.83276	−0.566759	−0.283379	+	0.959008i	$0.591456\pi$
$192$	0	0
$193$	−12.2056	−0.878575	−0.439287	−	0.898347i	$-0.644769\pi$
$193$	−12.2056	−0.878575	−0.439287	+	0.898347i	$0.644769\pi$
$194$	2.15165	0.154480
$195$	0	0
$196$	0	0
$197$	18.8222	1.34103	0.670513	−	0.741898i	$-0.266074\pi$
$197$	18.8222	1.34103	0.670513	+	0.741898i	$0.266074\pi$
$198$	0	0
$199$	−21.6116	−1.53201	−0.766004	−	0.642836i	$-0.777758\pi$
$199$	−21.6116	−1.53201	−0.766004	+	0.642836i	$0.777758\pi$
$200$	−3.75971	−0.265851
$201$	0	0
$202$	23.7633	1.67198
$203$	0	0
$204$	0	0
$205$	−11.8328	−0.826436
$206$	−8.00000	−0.557386
$207$	0	0
$208$	4.91638	0.340890
$209$	2.52444	0.174619
$210$	0	0
$211$	−17.3764	−1.19624	−0.598119	−	0.801407i	$-0.704085\pi$
$211$	−17.3764	−1.19624	−0.598119	+	0.801407i	$0.704085\pi$
$212$	3.21611	0.220884
$213$	0	0
$214$	−1.04888	−0.0716997
$215$	19.0192	1.29710
$216$	0	0
$217$	0	0
$218$	10.1078	0.684586
$219$	0	0
$220$	−11.2544	−0.758773
$221$	0.524438	0.0352775
$222$	0	0
$223$	10.5486	0.706388	0.353194	−	0.935550i	$-0.385096\pi$
$223$	10.5486	0.706388	0.353194	+	0.935550i	$0.385096\pi$
$224$	0	0
$225$	0	0
$226$	−9.87662	−0.656983
$227$	−6.95112	−0.461362	−0.230681	−	0.973029i	$-0.574095\pi$
$227$	−6.95112	−0.461362	−0.230681	+	0.973029i	$0.574095\pi$
$228$	0	0
$229$	21.0872	1.39348	0.696740	−	0.717323i	$-0.254633\pi$
$229$	21.0872	1.39348	0.696740	+	0.717323i	$0.254633\pi$
$230$	−37.4444	−2.46901
$231$	0	0
$232$	10.6861	0.701579
$233$	−6.08362	−0.398551	−0.199276	−	0.979943i	$-0.563859\pi$
$233$	−6.08362	−0.398551	−0.199276	+	0.979943i	$0.563859\pi$
$234$	0	0
$235$	−16.7980	−1.09578
$236$	−5.76328	−0.375157
$237$	0	0
$238$	0	0
$239$	14.2056	0.918881	0.459440	−	0.888209i	$-0.348050\pi$
$239$	14.2056	0.918881	0.459440	+	0.888209i	$0.348050\pi$
$240$	0	0
$241$	−8.44082	−0.543721	−0.271860	−	0.962337i	$-0.587639\pi$
$241$	−8.44082	−0.543721	−0.271860	+	0.962337i	$0.587639\pi$
$242$	−2.48970	−0.160044
$243$	0	0
$244$	2.57834	0.165061
$245$	0	0
$246$	0	0
$247$	0.813607	0.0517685
$248$	1.79445	0.113948
$249$	0	0
$250$	−10.6322	−0.672442
$251$	−23.9844	−1.51388	−0.756941	−	0.653483i	$-0.773307\pi$
$251$	−23.9844	−1.51388	−0.756941	+	0.653483i	$0.773307\pi$
$252$	0	0
$253$	22.7683	1.43143
$254$	−23.3622	−1.46588
$255$	0	0
$256$	20.8469	1.30293
$257$	15.6116	0.973827	0.486913	−	0.873450i	$-0.338123\pi$
$257$	15.6116	0.973827	0.486913	+	0.873450i	$0.338123\pi$
$258$	0	0
$259$	0	0
$260$	−3.62721	−0.224950
$261$	0	0
$262$	16.4111	1.01388
$263$	15.1708	0.935472	0.467736	−	0.883868i	$-0.345070\pi$
$263$	15.1708	0.935472	0.467736	+	0.883868i	$0.345070\pi$
$264$	0	0
$265$	7.01916	0.431183
$266$	0	0
$267$	0	0
$268$	12.9411	0.790502
$269$	23.2927	1.42018	0.710092	−	0.704109i	$-0.248653\pi$
$269$	23.2927	1.42018	0.710092	+	0.704109i	$0.248653\pi$
$270$	0	0
$271$	−12.7456	−0.774238	−0.387119	−	0.922030i	$-0.626530\pi$
$271$	−12.7456	−0.774238	−0.387119	+	0.922030i	$0.626530\pi$
$272$	2.57834	0.156335
$273$	0	0
$274$	11.3522	0.685810
$275$	−9.04888	−0.545668
$276$	0	0
$277$	8.12193	0.488000	0.244000	−	0.969775i	$-0.421540\pi$
$277$	8.12193	0.488000	0.244000	+	0.969775i	$0.421540\pi$
$278$	20.9894	1.25886
$279$	0	0
$280$	0	0
$281$	−19.0333	−1.13543	−0.567715	−	0.823225i	$-0.692173\pi$
$281$	−19.0333	−1.13543	−0.567715	+	0.823225i	$0.692173\pi$
$282$	0	0
$283$	−11.1466	−0.662598	−0.331299	−	0.943526i	$-0.607487\pi$
$283$	−11.1466	−0.662598	−0.331299	+	0.943526i	$0.607487\pi$
$284$	11.2544	0.667827
$285$	0	0
$286$	5.62721	0.332744
$287$	0	0
$288$	0	0
$289$	−16.7250	−0.983821
$290$	−42.2978	−2.48381
$291$	0	0
$292$	3.02061	0.176768
$293$	−14.1758	−0.828161	−0.414080	−	0.910240i	$-0.635897\pi$
$293$	−14.1758	−0.828161	−0.414080	+	0.910240i	$0.635897\pi$
$294$	0	0
$295$	−12.5783	−0.732339
$296$	7.93051	0.460952
$297$	0	0
$298$	−15.4600	−0.895572
$299$	7.33804	0.424370
$300$	0	0
$301$	0	0
$302$	−21.7350	−1.25071
$303$	0	0
$304$	4.00000	0.229416
$305$	5.62721	0.322213
$306$	0	0
$307$	−13.5592	−0.773863	−0.386932	−	0.922108i	$-0.626465\pi$
$307$	−13.5592	−0.773863	−0.386932	+	0.922108i	$0.626465\pi$
$308$	0	0
$309$	0	0
$310$	−7.10278	−0.403411
$311$	0.426686	0.0241952	0.0120976	−	0.999927i	$-0.496149\pi$
$311$	0.426686	0.0241952	0.0120976	+	0.999927i	$0.496149\pi$
$312$	0	0
$313$	18.1517	1.02599	0.512996	−	0.858391i	$-0.328536\pi$
$313$	18.1517	1.02599	0.512996	+	0.858391i	$0.328536\pi$
$314$	23.2645	1.31289
$315$	0	0
$316$	−17.4600	−0.982200
$317$	−9.42166	−0.529173	−0.264587	−	0.964362i	$-0.585236\pi$
$317$	−9.42166	−0.529173	−0.264587	+	0.964362i	$0.585236\pi$
$318$	0	0
$319$	25.7194	1.44001
$320$	−4.67609	−0.261401
$321$	0	0
$322$	0	0
$323$	0.426686	0.0237415
$324$	0	0
$325$	−2.91638	−0.161772
$326$	−24.4111	−1.35201
$327$	0	0
$328$	5.42166	0.299361
$329$	0	0
$330$	0	0
$331$	−17.4005	−0.956420	−0.478210	−	0.878246i	$-0.658714\pi$
$331$	−17.4005	−0.956420	−0.478210	+	0.878246i	$0.658714\pi$
$332$	21.2444	1.16594
$333$	0	0
$334$	−3.68111	−0.201421
$335$	28.2439	1.54313
$336$	0	0
$337$	22.0524	1.20127	0.600637	−	0.799522i	$-0.294914\pi$
$337$	22.0524	1.20127	0.600637	+	0.799522i	$0.294914\pi$
$338$	1.81361	0.0986472
$339$	0	0
$340$	−1.90225	−0.103164
$341$	4.31889	0.233881
$342$	0	0
$343$	0	0
$344$	−8.71440	−0.469849
$345$	0	0
$346$	36.8122	1.97903
$347$	−25.3522	−1.36098	−0.680488	−	0.732759i	$-0.738232\pi$
$347$	−25.3522	−1.36098	−0.680488	+	0.732759i	$0.738232\pi$
$348$	0	0
$349$	5.70529	0.305397	0.152699	−	0.988273i	$-0.451204\pi$
$349$	5.70529	0.305397	0.152699	+	0.988273i	$0.451204\pi$
$350$	0	0
$351$	0	0
$352$	19.6655	1.04818
$353$	−28.6761	−1.52627	−0.763137	−	0.646237i	$-0.776342\pi$
$353$	−28.6761	−1.52627	−0.763137	+	0.646237i	$0.776342\pi$
$354$	0	0
$355$	24.5628	1.30366
$356$	−13.7250	−0.727422
$357$	0	0
$358$	19.9561	1.05472
$359$	−11.0433	−0.582845	−0.291423	−	0.956594i	$-0.594129\pi$
$359$	−11.0433	−0.582845	−0.291423	+	0.956594i	$0.594129\pi$
$360$	0	0
$361$	−18.3380	−0.965160
$362$	−1.25443	−0.0659312
$363$	0	0
$364$	0	0
$365$	6.59247	0.345066
$366$	0	0
$367$	−27.3466	−1.42748	−0.713741	−	0.700409i	$-0.753001\pi$
$367$	−27.3466	−1.42748	−0.713741	+	0.700409i	$0.753001\pi$
$368$	36.0766	1.88062
$369$	0	0
$370$	−31.3905	−1.63191
$371$	0	0
$372$	0	0
$373$	16.1461	0.836014	0.418007	−	0.908444i	$-0.362729\pi$
$373$	16.1461	0.836014	0.418007	+	0.908444i	$0.362729\pi$
$374$	2.95112	0.152599
$375$	0	0
$376$	7.69670	0.396927
$377$	8.28917	0.426914
$378$	0	0
$379$	26.1305	1.34223	0.671117	−	0.741351i	$-0.265815\pi$
$379$	26.1305	1.34223	0.671117	+	0.741351i	$0.265815\pi$
$380$	−2.95112	−0.151389
$381$	0	0
$382$	−14.2056	−0.726819
$383$	−21.0489	−1.07555	−0.537774	−	0.843089i	$-0.680735\pi$
$383$	−21.0489	−1.07555	−0.537774	+	0.843089i	$0.680735\pi$
$384$	0	0
$385$	0	0
$386$	−22.1361	−1.12670
$387$	0	0
$388$	1.52946	0.0776466
$389$	21.6061	1.09547	0.547736	−	0.836651i	$-0.315490\pi$
$389$	21.6061	1.09547	0.547736	+	0.836651i	$0.315490\pi$
$390$	0	0
$391$	3.84835	0.194619
$392$	0	0
$393$	0	0
$394$	34.1361	1.71975
$395$	−38.1063	−1.91734
$396$	0	0
$397$	27.6952	1.38998	0.694992	−	0.719017i	$-0.255408\pi$
$397$	27.6952	1.38998	0.694992	+	0.719017i	$0.255408\pi$
$398$	−39.1950	−1.96467
$399$	0	0
$400$	−14.3380	−0.716902
$401$	2.57834	0.128756	0.0643780	−	0.997926i	$-0.479494\pi$
$401$	2.57834	0.128756	0.0643780	+	0.997926i	$0.479494\pi$
$402$	0	0
$403$	1.39194	0.0693376
$404$	16.8917	0.840393
$405$	0	0
$406$	0	0
$407$	19.0872	0.946117
$408$	0	0
$409$	−15.1169	−0.747483	−0.373742	−	0.927533i	$-0.621925\pi$
$409$	−15.1169	−0.747483	−0.373742	+	0.927533i	$0.621925\pi$
$410$	−21.4600	−1.05983
$411$	0	0
$412$	−5.68665	−0.280161
$413$	0	0
$414$	0	0
$415$	46.3658	2.27601
$416$	6.33804	0.310748
$417$	0	0
$418$	4.57834	0.223934
$419$	−9.99446	−0.488261	−0.244131	−	0.969742i	$-0.578503\pi$
$419$	−9.99446	−0.488261	−0.244131	+	0.969742i	$0.578503\pi$
$420$	0	0
$421$	−25.9250	−1.26351	−0.631753	−	0.775170i	$-0.717664\pi$
$421$	−25.9250	−1.26351	−0.631753	+	0.775170i	$0.717664\pi$
$422$	−31.5139	−1.53407
$423$	0	0
$424$	−3.21611	−0.156188
$425$	−1.52946	−0.0741898
$426$	0	0
$427$	0	0
$428$	−0.745574	−0.0360387
$429$	0	0
$430$	34.4933	1.66341
$431$	−30.6761	−1.47762	−0.738808	−	0.673916i	$-0.764611\pi$
$431$	−30.6761	−1.47762	−0.738808	+	0.673916i	$0.764611\pi$
$432$	0	0
$433$	3.51941	0.169132	0.0845661	−	0.996418i	$-0.473050\pi$
$433$	3.51941	0.169132	0.0845661	+	0.996418i	$0.473050\pi$
$434$	0	0
$435$	0	0
$436$	7.18494	0.344096
$437$	5.97028	0.285597
$438$	0	0
$439$	−32.3517	−1.54406	−0.772030	−	0.635586i	$-0.780759\pi$
$439$	−32.3517	−1.54406	−0.772030	+	0.635586i	$0.780759\pi$
$440$	11.2544	0.536534
$441$	0	0
$442$	0.951124	0.0452404
$443$	−15.4458	−0.733854	−0.366927	−	0.930250i	$-0.619590\pi$
$443$	−15.4458	−0.733854	−0.366927	+	0.930250i	$0.619590\pi$
$444$	0	0
$445$	−29.9547	−1.41999
$446$	19.1310	0.905881
$447$	0	0
$448$	0	0
$449$	14.4705	0.682907	0.341453	−	0.939899i	$-0.389081\pi$
$449$	14.4705	0.682907	0.341453	+	0.939899i	$0.389081\pi$
$450$	0	0
$451$	13.0489	0.614448
$452$	−7.02061	−0.330222
$453$	0	0
$454$	−12.6066	−0.591657
$455$	0	0
$456$	0	0
$457$	−34.6705	−1.62182	−0.810910	−	0.585171i	$-0.801027\pi$
$457$	−34.6705	−1.62182	−0.810910	+	0.585171i	$0.801027\pi$
$458$	38.2439	1.78702
$459$	0	0
$460$	−26.6167	−1.24101
$461$	12.5400	0.584047	0.292024	−	0.956411i	$-0.405671\pi$
$461$	12.5400	0.584047	0.292024	+	0.956411i	$0.405671\pi$
$462$	0	0
$463$	12.1517	0.564735	0.282368	−	0.959306i	$-0.408880\pi$
$463$	12.1517	0.564735	0.282368	+	0.959306i	$0.408880\pi$
$464$	40.7527	1.89190
$465$	0	0
$466$	−11.0333	−0.511107
$467$	−37.0333	−1.71370	−0.856848	−	0.515569i	$-0.827581\pi$
$467$	−37.0333	−1.71370	−0.856848	+	0.515569i	$0.827581\pi$
$468$	0	0
$469$	0	0
$470$	−30.4650	−1.40525
$471$	0	0
$472$	5.76328	0.265276
$473$	−20.9739	−0.964379
$474$	0	0
$475$	−2.37279	−0.108871
$476$	0	0
$477$	0	0
$478$	25.7633	1.17838
$479$	12.0086	0.548687	0.274343	−	0.961632i	$-0.411540\pi$
$479$	12.0086	0.548687	0.274343	+	0.961632i	$0.411540\pi$
$480$	0	0
$481$	6.15165	0.280491
$482$	−15.3083	−0.697275
$483$	0	0
$484$	−1.76975	−0.0804434
$485$	3.33804	0.151573
$486$	0	0
$487$	11.1184	0.503821	0.251911	−	0.967751i	$-0.418941\pi$
$487$	11.1184	0.503821	0.251911	+	0.967751i	$0.418941\pi$
$488$	−2.57834	−0.116716
$489$	0	0
$490$	0	0
$491$	−0.0594386	−0.00268243	−0.00134121	−	0.999999i	$-0.500427\pi$
$491$	−0.0594386	−0.00268243	−0.00134121	+	0.999999i	$0.500427\pi$
$492$	0	0
$493$	4.34715	0.195786
$494$	1.47556	0.0663887
$495$	0	0
$496$	6.84333	0.307274
$497$	0	0
$498$	0	0
$499$	10.2978	0.460991	0.230496	−	0.973073i	$-0.425965\pi$
$499$	10.2978	0.460991	0.230496	+	0.973073i	$0.425965\pi$
$500$	−7.55773	−0.337992
$501$	0	0
$502$	−43.4983	−1.94142
$503$	−9.32391	−0.415733	−0.207866	−	0.978157i	$-0.566652\pi$
$503$	−9.32391	−0.415733	−0.207866	+	0.978157i	$0.566652\pi$
$504$	0	0
$505$	36.8661	1.64052
$506$	41.2927	1.83569
$507$	0	0
$508$	−16.6066	−0.736799
$509$	39.6952	1.75946	0.879730	−	0.475473i	$-0.157723\pi$
$509$	39.6952	1.75946	0.879730	+	0.475473i	$0.157723\pi$
$510$	0	0
$511$	0	0
$512$	18.4842	0.816892
$513$	0	0
$514$	28.3133	1.24885
$515$	−12.4111	−0.546898
$516$	0	0
$517$	18.5244	0.814704
$518$	0	0
$519$	0	0
$520$	3.62721	0.159064
$521$	22.3627	0.979729	0.489865	−	0.871798i	$-0.337046\pi$
$521$	22.3627	0.979729	0.489865	+	0.871798i	$0.337046\pi$
$522$	0	0
$523$	20.6550	0.903178	0.451589	−	0.892226i	$-0.350857\pi$
$523$	20.6550	0.903178	0.451589	+	0.892226i	$0.350857\pi$
$524$	11.6655	0.509611
$525$	0	0
$526$	27.5139	1.19966
$527$	0.729988	0.0317988
$528$	0	0
$529$	30.8469	1.34117
$530$	12.7300	0.552955
$531$	0	0
$532$	0	0
$533$	4.20555	0.182163
$534$	0	0
$535$	−1.62721	−0.0703506
$536$	−12.9411	−0.558969
$537$	0	0
$538$	42.2439	1.82126
$539$	0	0
$540$	0	0
$541$	5.62167	0.241695	0.120847	−	0.992671i	$-0.461439\pi$
$541$	5.62167	0.241695	0.120847	+	0.992671i	$0.461439\pi$
$542$	−23.1155	−0.992894
$543$	0	0
$544$	3.32391	0.142512
$545$	15.6811	0.671705
$546$	0	0
$547$	−10.3970	−0.444542	−0.222271	−	0.974985i	$-0.571347\pi$
$547$	−10.3970	−0.444542	−0.222271	+	0.974985i	$0.571347\pi$
$548$	8.06949	0.344711
$549$	0	0
$550$	−16.4111	−0.699772
$551$	6.74412	0.287309
$552$	0	0
$553$	0	0
$554$	14.7300	0.625817
$555$	0	0
$556$	14.9200	0.632747
$557$	−14.6550	−0.620951	−0.310475	−	0.950581i	$-0.600488\pi$
$557$	−14.6550	−0.620951	−0.310475	+	0.950581i	$0.600488\pi$
$558$	0	0
$559$	−6.75971	−0.285905
$560$	0	0
$561$	0	0
$562$	−34.5189	−1.45609
$563$	−24.7456	−1.04290	−0.521451	−	0.853281i	$-0.674609\pi$
$563$	−24.7456	−1.04290	−0.521451	+	0.853281i	$0.674609\pi$
$564$	0	0
$565$	−15.3225	−0.644621
$566$	−20.2156	−0.849725
$567$	0	0
$568$	−11.2544	−0.472225
$569$	20.5330	0.860789	0.430395	−	0.902641i	$-0.358374\pi$
$569$	20.5330	0.860789	0.430395	+	0.902641i	$0.358374\pi$
$570$	0	0
$571$	−41.8953	−1.75326	−0.876631	−	0.481163i	$-0.840214\pi$
$571$	−41.8953	−1.75326	−0.876631	+	0.481163i	$0.840214\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	0	0
$575$	−21.4005	−0.892464
$576$	0	0
$577$	20.1744	0.839870	0.419935	−	0.907554i	$-0.362053\pi$
$577$	20.1744	0.839870	0.419935	+	0.907554i	$0.362053\pi$
$578$	−30.3325	−1.26167
$579$	0	0
$580$	−30.0666	−1.24845
$581$	0	0
$582$	0	0
$583$	−7.74055	−0.320581
$584$	−3.02061	−0.124994
$585$	0	0
$586$	−25.7094	−1.06204
$587$	−18.7441	−0.773653	−0.386826	−	0.922153i	$-0.626429\pi$
$587$	−18.7441	−0.773653	−0.386826	+	0.922153i	$0.626429\pi$
$588$	0	0
$589$	1.13249	0.0466636
$590$	−22.8122	−0.939162
$591$	0	0
$592$	30.2439	1.24302
$593$	−2.98084	−0.122409	−0.0612043	−	0.998125i	$-0.519494\pi$
$593$	−2.98084	−0.122409	−0.0612043	+	0.998125i	$0.519494\pi$
$594$	0	0
$595$	0	0
$596$	−10.9894	−0.450145
$597$	0	0
$598$	13.3083	0.544218
$599$	−7.47411	−0.305384	−0.152692	−	0.988274i	$-0.548794\pi$
$599$	−7.47411	−0.305384	−0.152692	+	0.988274i	$0.548794\pi$
$600$	0	0
$601$	−21.4700	−0.875780	−0.437890	−	0.899028i	$-0.644274\pi$
$601$	−21.4700	−0.875780	−0.437890	+	0.899028i	$0.644274\pi$
$602$	0	0
$603$	0	0
$604$	−15.4499	−0.628649
$605$	−3.86248	−0.157032
$606$	0	0
$607$	22.9044	0.929660	0.464830	−	0.885400i	$-0.346116\pi$
$607$	22.9044	0.929660	0.464830	+	0.885400i	$0.346116\pi$
$608$	5.15667	0.209131
$609$	0	0
$610$	10.2056	0.413211
$611$	5.97028	0.241532
$612$	0	0
$613$	−20.1461	−0.813694	−0.406847	−	0.913496i	$-0.633372\pi$
$613$	−20.1461	−0.813694	−0.406847	+	0.913496i	$0.633372\pi$
$614$	−24.5910	−0.992413
$615$	0	0
$616$	0	0
$617$	13.7844	0.554939	0.277470	−	0.960734i	$-0.410504\pi$
$617$	13.7844	0.554939	0.277470	+	0.960734i	$0.410504\pi$
$618$	0	0
$619$	19.6655	0.790424	0.395212	−	0.918590i	$-0.370671\pi$
$619$	19.6655	0.790424	0.395212	+	0.918590i	$0.370671\pi$
$620$	−5.04888	−0.202768
$621$	0	0
$622$	0.773841	0.0310282
$623$	0	0
$624$	0	0
$625$	−31.0766	−1.24307
$626$	32.9200	1.31575
$627$	0	0
$628$	16.5371	0.659903
$629$	3.22616	0.128635
$630$	0	0
$631$	−16.1672	−0.643608	−0.321804	−	0.946806i	$-0.604289\pi$
$631$	−16.1672	−0.643608	−0.321804	+	0.946806i	$0.604289\pi$
$632$	17.4600	0.694521
$633$	0	0
$634$	−17.0872	−0.678619
$635$	−36.2439	−1.43829
$636$	0	0
$637$	0	0
$638$	46.6449	1.84669
$639$	0	0
$640$	27.1849	1.07458
$641$	29.0036	1.14557	0.572786	−	0.819705i	$-0.305863\pi$
$641$	29.0036	1.14557	0.572786	+	0.819705i	$0.305863\pi$
$642$	0	0
$643$	39.2233	1.54681	0.773407	−	0.633910i	$-0.218551\pi$
$643$	39.2233	1.54681	0.773407	+	0.633910i	$0.218551\pi$
$644$	0	0
$645$	0	0
$646$	0.773841	0.0304464
$647$	−11.9844	−0.471156	−0.235578	−	0.971855i	$-0.575698\pi$
$647$	−11.9844	−0.471156	−0.235578	+	0.971855i	$0.575698\pi$
$648$	0	0
$649$	13.8711	0.544487
$650$	−5.28917	−0.207458
$651$	0	0
$652$	−17.3522	−0.679564
$653$	−45.3311	−1.77394	−0.886971	−	0.461826i	$-0.847194\pi$
$653$	−45.3311	−1.77394	−0.886971	+	0.461826i	$0.847194\pi$
$654$	0	0
$655$	25.4600	0.994804
$656$	20.6761	0.807266
$657$	0	0
$658$	0	0
$659$	−6.12193	−0.238477	−0.119238	−	0.992866i	$-0.538045\pi$
$659$	−6.12193	−0.238477	−0.119238	+	0.992866i	$0.538045\pi$
$660$	0	0
$661$	27.5280	1.07072	0.535358	−	0.844625i	$-0.320177\pi$
$661$	27.5280	1.07072	0.535358	+	0.844625i	$0.320177\pi$
$662$	−31.5577	−1.22653
$663$	0	0
$664$	−21.2444	−0.824442
$665$	0	0
$666$	0	0
$667$	60.8263	2.35520
$668$	−2.61665	−0.101241
$669$	0	0
$670$	51.2233	1.97893
$671$	−6.20555	−0.239563
$672$	0	0
$673$	27.9547	1.07757	0.538787	−	0.842442i	$-0.318883\pi$
$673$	27.9547	1.07757	0.538787	+	0.842442i	$0.318883\pi$
$674$	39.9945	1.54053
$675$	0	0
$676$	1.28917	0.0495834
$677$	−12.6605	−0.486583	−0.243291	−	0.969953i	$-0.578227\pi$
$677$	−12.6605	−0.486583	−0.243291	+	0.969953i	$0.578227\pi$
$678$	0	0
$679$	0	0
$680$	1.90225	0.0729479
$681$	0	0
$682$	7.83276	0.299932
$683$	28.3033	1.08300	0.541498	−	0.840702i	$-0.317857\pi$
$683$	28.3033	1.08300	0.541498	+	0.840702i	$0.317857\pi$
$684$	0	0
$685$	17.6116	0.672906
$686$	0	0
$687$	0	0
$688$	−33.2333	−1.26701
$689$	−2.49472	−0.0950412
$690$	0	0
$691$	−12.2353	−0.465452	−0.232726	−	0.972542i	$-0.574764\pi$
$691$	−12.2353	−0.465452	−0.232726	+	0.972542i	$0.574764\pi$
$692$	26.1672	0.994729
$693$	0	0
$694$	−45.9789	−1.74533
$695$	32.5628	1.23518
$696$	0	0
$697$	2.20555	0.0835412
$698$	10.3472	0.391646
$699$	0	0
$700$	0	0
$701$	−51.0419	−1.92783	−0.963913	−	0.266219i	$-0.914226\pi$
$701$	−51.0419	−1.92783	−0.963913	+	0.266219i	$0.914226\pi$
$702$	0	0
$703$	5.00502	0.188768
$704$	5.15667	0.194349
$705$	0	0
$706$	−52.0071	−1.95731
$707$	0	0
$708$	0	0
$709$	42.5910	1.59954	0.799770	−	0.600307i	$-0.204955\pi$
$709$	42.5910	1.59954	0.799770	+	0.600307i	$0.204955\pi$
$710$	44.5472	1.67183
$711$	0	0
$712$	13.7250	0.514365
$713$	10.2141	0.382523
$714$	0	0
$715$	8.72999	0.326483
$716$	14.1855	0.530135
$717$	0	0
$718$	−20.0283	−0.747448
$719$	−42.4933	−1.58473	−0.792366	−	0.610046i	$-0.791151\pi$
$719$	−42.4933	−1.58473	−0.792366	+	0.610046i	$0.791151\pi$
$720$	0	0
$721$	0	0
$722$	−33.2580	−1.23773
$723$	0	0
$724$	−0.891685	−0.0331392
$725$	−24.1744	−0.897814
$726$	0	0
$727$	3.75614	0.139307	0.0696537	−	0.997571i	$-0.477811\pi$
$727$	3.75614	0.139307	0.0696537	+	0.997571i	$0.477811\pi$
$728$	0	0
$729$	0	0
$730$	11.9561	0.442517
$731$	−3.54505	−0.131118
$732$	0	0
$733$	−45.7819	−1.69099	−0.845497	−	0.533980i	$-0.820696\pi$
$733$	−45.7819	−1.69099	−0.845497	+	0.533980i	$0.820696\pi$
$734$	−49.5960	−1.83062
$735$	0	0
$736$	46.5089	1.71434
$737$	−31.1466	−1.14730
$738$	0	0
$739$	−14.0539	−0.516981	−0.258491	−	0.966014i	$-0.583225\pi$
$739$	−14.0539	−0.516981	−0.258491	+	0.966014i	$0.583225\pi$
$740$	−22.3133	−0.820255
$741$	0	0
$742$	0	0
$743$	−4.74557	−0.174098	−0.0870491	−	0.996204i	$-0.527744\pi$
$743$	−4.74557	−0.174098	−0.0870491	+	0.996204i	$0.527744\pi$
$744$	0	0
$745$	−23.9844	−0.878721
$746$	29.2827	1.07212
$747$	0	0
$748$	2.09775	0.0767014
$749$	0	0
$750$	0	0
$751$	36.1008	1.31734	0.658669	−	0.752433i	$-0.271120\pi$
$751$	36.1008	1.31734	0.658669	+	0.752433i	$0.271120\pi$
$752$	29.3522	1.07036
$753$	0	0
$754$	15.0333	0.547480
$755$	−33.7194	−1.22718
$756$	0	0
$757$	−1.03474	−0.0376084	−0.0188042	−	0.999823i	$-0.505986\pi$
$757$	−1.03474	−0.0376084	−0.0188042	+	0.999823i	$0.505986\pi$
$758$	47.3905	1.72130
$759$	0	0
$760$	2.95112	0.107049
$761$	29.8414	1.08175	0.540874	−	0.841104i	$-0.318094\pi$
$761$	29.8414	1.08175	0.540874	+	0.841104i	$0.318094\pi$
$762$	0	0
$763$	0	0
$764$	−10.0978	−0.365324
$765$	0	0
$766$	−38.1744	−1.37930
$767$	4.47054	0.161422
$768$	0	0
$769$	−23.6358	−0.852329	−0.426164	−	0.904646i	$-0.640136\pi$
$769$	−23.6358	−0.852329	−0.426164	+	0.904646i	$0.640136\pi$
$770$	0	0
$771$	0	0
$772$	−15.7350	−0.566315
$773$	13.0278	0.468576	0.234288	−	0.972167i	$-0.424724\pi$
$773$	13.0278	0.468576	0.234288	+	0.972167i	$0.424724\pi$
$774$	0	0
$775$	−4.05944	−0.145819
$776$	−1.52946	−0.0549045
$777$	0	0
$778$	39.1849	1.40485
$779$	3.42166	0.122594
$780$	0	0
$781$	−27.0872	−0.969256
$782$	6.97939	0.249583
$783$	0	0
$784$	0	0
$785$	36.0922	1.28819
$786$	0	0
$787$	−46.2141	−1.64736	−0.823678	−	0.567058i	$-0.808082\pi$
$787$	−46.2141	−1.64736	−0.823678	+	0.567058i	$0.808082\pi$
$788$	24.2650	0.864404
$789$	0	0
$790$	−69.1099	−2.45882
$791$	0	0
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	50.2283	1.78253
$795$	0	0
$796$	−27.8610	−0.987508
$797$	53.1155	1.88145	0.940723	−	0.339176i	$-0.110148\pi$
$797$	53.1155	1.88145	0.940723	+	0.339176i	$0.110148\pi$
$798$	0	0
$799$	3.13104	0.110768
$800$	−18.4842	−0.653514
$801$	0	0
$802$	4.67609	0.165118
$803$	−7.27001	−0.256553
$804$	0	0
$805$	0	0
$806$	2.52444	0.0889195
$807$	0	0
$808$	−16.8917	−0.594247
$809$	54.4635	1.91484	0.957418	−	0.288705i	$-0.0932246\pi$
$809$	54.4635	1.91484	0.957418	+	0.288705i	$0.0932246\pi$
$810$	0	0
$811$	−38.0978	−1.33779	−0.668897	−	0.743356i	$-0.733233\pi$
$811$	−38.0978	−1.33779	−0.668897	+	0.743356i	$0.733233\pi$
$812$	0	0
$813$	0	0
$814$	34.6167	1.21331
$815$	−37.8711	−1.32657
$816$	0	0
$817$	−5.49974	−0.192412
$818$	−27.4161	−0.958582
$819$	0	0
$820$	−15.2544	−0.532708
$821$	2.30330	0.0803858	0.0401929	−	0.999192i	$-0.487203\pi$
$821$	2.30330	0.0803858	0.0401929	+	0.999192i	$0.487203\pi$
$822$	0	0
$823$	23.6172	0.823243	0.411621	−	0.911355i	$-0.364963\pi$
$823$	23.6172	0.823243	0.411621	+	0.911355i	$0.364963\pi$
$824$	5.68665	0.198104
$825$	0	0
$826$	0	0
$827$	48.1643	1.67484	0.837419	−	0.546562i	$-0.184064\pi$
$827$	48.1643	1.67484	0.837419	+	0.546562i	$0.184064\pi$
$828$	0	0
$829$	−13.0716	−0.453996	−0.226998	−	0.973895i	$-0.572891\pi$
$829$	−13.0716	−0.453996	−0.226998	+	0.973895i	$0.572891\pi$
$830$	84.0893	2.91878
$831$	0	0
$832$	1.66196	0.0576179
$833$	0	0
$834$	0	0
$835$	−5.71083	−0.197631
$836$	3.25443	0.112557
$837$	0	0
$838$	−18.1260	−0.626153
$839$	−17.6756	−0.610229	−0.305114	−	0.952316i	$-0.598695\pi$
$839$	−17.6756	−0.610229	−0.305114	+	0.952316i	$0.598695\pi$
$840$	0	0
$841$	39.7103	1.36932
$842$	−47.0177	−1.62034
$843$	0	0
$844$	−22.4011	−0.771076
$845$	2.81361	0.0967910
$846$	0	0
$847$	0	0
$848$	−12.2650	−0.421181
$849$	0	0
$850$	−2.77384	−0.0951420
$851$	45.1411	1.54742
$852$	0	0
$853$	5.48970	0.187964	0.0939818	−	0.995574i	$-0.470040\pi$
$853$	5.48970	0.187964	0.0939818	+	0.995574i	$0.470040\pi$
$854$	0	0
$855$	0	0
$856$	0.745574	0.0254832
$857$	−11.0489	−0.377422	−0.188711	−	0.982033i	$-0.560431\pi$
$857$	−11.0489	−0.377422	−0.188711	+	0.982033i	$0.560431\pi$
$858$	0	0
$859$	−45.2616	−1.54430	−0.772152	−	0.635437i	$-0.780820\pi$
$859$	−45.2616	−1.54430	−0.772152	+	0.635437i	$0.780820\pi$
$860$	24.5189	0.836087
$861$	0	0
$862$	−55.6344	−1.89491
$863$	5.90225	0.200915	0.100457	−	0.994941i	$-0.467969\pi$
$863$	5.90225	0.200915	0.100457	+	0.994941i	$0.467969\pi$
$864$	0	0
$865$	57.1099	1.94180
$866$	6.38283	0.216898
$867$	0	0
$868$	0	0
$869$	42.0227	1.42552
$870$	0	0
$871$	−10.0383	−0.340135
$872$	−7.18494	−0.243313
$873$	0	0
$874$	10.8277	0.366254
$875$	0	0
$876$	0	0
$877$	−4.90727	−0.165707	−0.0828534	−	0.996562i	$-0.526403\pi$
$877$	−4.90727	−0.165707	−0.0828534	+	0.996562i	$0.526403\pi$
$878$	−58.6732	−1.98012
$879$	0	0
$880$	42.9200	1.44683
$881$	44.2822	1.49190	0.745952	−	0.665999i	$-0.231995\pi$
$881$	44.2822	1.49190	0.745952	+	0.665999i	$0.231995\pi$
$882$	0	0
$883$	−58.8605	−1.98081	−0.990407	−	0.138181i	$-0.955874\pi$
$883$	−58.8605	−1.98081	−0.990407	+	0.138181i	$0.955874\pi$
$884$	0.676089	0.0227393
$885$	0	0
$886$	−28.0127	−0.941104
$887$	−10.1289	−0.340096	−0.170048	−	0.985436i	$-0.554392\pi$
$887$	−10.1289	−0.340096	−0.170048	+	0.985436i	$0.554392\pi$
$888$	0	0
$889$	0	0
$890$	−54.3260	−1.82101
$891$	0	0
$892$	13.5989	0.455326
$893$	4.85746	0.162549
$894$	0	0
$895$	30.9597	1.03487
$896$	0	0
$897$	0	0
$898$	26.2439	0.875769
$899$	11.5381	0.384816
$900$	0	0
$901$	−1.30833	−0.0435866
$902$	23.6655	0.787976
$903$	0	0
$904$	7.02061	0.233502
$905$	−1.94610	−0.0646906
$906$	0	0
$907$	37.9547	1.26026	0.630132	−	0.776488i	$-0.283001\pi$
$907$	37.9547	1.26026	0.630132	+	0.776488i	$0.283001\pi$
$908$	−8.96117	−0.297387
$909$	0	0
$910$	0	0
$911$	−5.57477	−0.184700	−0.0923501	−	0.995727i	$-0.529438\pi$
$911$	−5.57477	−0.184700	−0.0923501	+	0.995727i	$0.529438\pi$
$912$	0	0
$913$	−51.1310	−1.69219
$914$	−62.8787	−2.07984
$915$	0	0
$916$	27.1849	0.898216
$917$	0	0
$918$	0	0
$919$	15.7844	0.520679	0.260340	−	0.965517i	$-0.416165\pi$
$919$	15.7844	0.520679	0.260340	+	0.965517i	$0.416165\pi$
$920$	26.6167	0.877525
$921$	0	0
$922$	22.7427	0.748990
$923$	−8.72999	−0.287351
$924$	0	0
$925$	−17.9406	−0.589882
$926$	22.0383	0.724224
$927$	0	0
$928$	52.5371	1.72462
$929$	−45.2630	−1.48503	−0.742516	−	0.669829i	$-0.766368\pi$
$929$	−45.2630	−1.48503	−0.742516	+	0.669829i	$0.766368\pi$
$930$	0	0
$931$	0	0
$932$	−7.84281	−0.256900
$933$	0	0
$934$	−67.1638	−2.19767
$935$	4.57834	0.149728
$936$	0	0
$937$	−53.6188	−1.75165	−0.875824	−	0.482630i	$-0.839682\pi$
$937$	−53.6188	−1.75165	−0.875824	+	0.482630i	$0.839682\pi$
$938$	0	0
$939$	0	0
$940$	−21.6555	−0.706324
$941$	20.7753	0.677255	0.338628	−	0.940920i	$-0.390037\pi$
$941$	20.7753	0.677255	0.338628	+	0.940920i	$0.390037\pi$
$942$	0	0
$943$	30.8605	1.00496
$944$	21.9789	0.715351
$945$	0	0
$946$	−38.0383	−1.23673
$947$	10.8605	0.352919	0.176460	−	0.984308i	$-0.443535\pi$
$947$	10.8605	0.352919	0.176460	+	0.984308i	$0.443535\pi$
$948$	0	0
$949$	−2.34307	−0.0760592
$950$	−4.30330	−0.139618
$951$	0	0
$952$	0	0
$953$	25.7180	0.833087	0.416543	−	0.909116i	$-0.363241\pi$
$953$	25.7180	0.833087	0.416543	+	0.909116i	$0.363241\pi$
$954$	0	0
$955$	−22.0383	−0.713143
$956$	18.3133	0.592296
$957$	0	0
$958$	21.7789	0.703643
$959$	0	0
$960$	0	0
$961$	−29.0625	−0.937500
$962$	11.1567	0.359706
$963$	0	0
$964$	−10.8816	−0.350474
$965$	−34.3416	−1.10550
$966$	0	0
$967$	33.5038	1.07741	0.538705	−	0.842494i	$-0.318914\pi$
$967$	33.5038	1.07741	0.538705	+	0.842494i	$0.318914\pi$
$968$	1.76975	0.0568820
$969$	0	0
$970$	6.05390	0.194379
$971$	2.03831	0.0654126	0.0327063	−	0.999465i	$-0.489587\pi$
$971$	2.03831	0.0654126	0.0327063	+	0.999465i	$0.489587\pi$
$972$	0	0
$973$	0	0
$974$	20.1643	0.646107
$975$	0	0
$976$	−9.83276	−0.314739
$977$	−15.1411	−0.484406	−0.242203	−	0.970226i	$-0.577870\pi$
$977$	−15.1411	−0.484406	−0.242203	+	0.970226i	$0.577870\pi$
$978$	0	0
$979$	33.0333	1.05575
$980$	0	0
$981$	0	0
$982$	−0.107798	−0.00343998
$983$	49.3124	1.57282	0.786411	−	0.617704i	$-0.211937\pi$
$983$	49.3124	1.57282	0.786411	+	0.617704i	$0.211937\pi$
$984$	0	0
$985$	52.9583	1.68739
$986$	7.88403	0.251079
$987$	0	0
$988$	1.04888	0.0333692
$989$	−49.6030	−1.57728
$990$	0	0
$991$	5.43171	0.172544	0.0862720	−	0.996272i	$-0.472505\pi$
$991$	5.43171	0.172544	0.0862720	+	0.996272i	$0.472505\pi$
$992$	8.82220	0.280105
$993$	0	0
$994$	0	0
$995$	−60.8066	−1.92770
$996$	0	0
$997$	53.6061	1.69772	0.848861	−	0.528616i	$-0.177289\pi$
$997$	53.6061	1.69772	0.848861	+	0.528616i	$0.177289\pi$
$998$	18.6761	0.591181
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.x.1.3		3
3.2	odd	2		637.2.a.j.1.1		3
7.6	odd	2		819.2.a.i.1.3		3
21.2	odd	6		637.2.e.i.508.3		6
21.5	even	6		637.2.e.j.508.3		6
21.11	odd	6		637.2.e.i.79.3		6
21.17	even	6		637.2.e.j.79.3		6
21.20	even	2		91.2.a.d.1.1	✓	3
39.38	odd	2		8281.2.a.bg.1.3		3
84.83	odd	2		1456.2.a.t.1.3		3
105.104	even	2		2275.2.a.m.1.3		3
168.83	odd	2		5824.2.a.bs.1.1		3
168.125	even	2		5824.2.a.by.1.3		3
273.83	odd	4		1183.2.c.f.337.5		6
273.125	odd	4		1183.2.c.f.337.2		6
273.272	even	2		1183.2.a.i.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.d.1.1	✓	3	21.20	even	2
637.2.a.j.1.1		3	3.2	odd	2
637.2.e.i.79.3		6	21.11	odd	6
637.2.e.i.508.3		6	21.2	odd	6
637.2.e.j.79.3		6	21.17	even	6
637.2.e.j.508.3		6	21.5	even	6
819.2.a.i.1.3		3	7.6	odd	2
1183.2.a.i.1.3		3	273.272	even	2
1183.2.c.f.337.2		6	273.125	odd	4
1183.2.c.f.337.5		6	273.83	odd	4
1456.2.a.t.1.3		3	84.83	odd	2
2275.2.a.m.1.3		3	105.104	even	2
5733.2.a.x.1.3		3	1.1	even	1	trivial
5824.2.a.bs.1.1		3	168.83	odd	2
5824.2.a.by.1.3		3	168.125	even	2
8281.2.a.bg.1.3		3	39.38	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:	\( N \)	\(=\)	\( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5733.a (trivial)

\(f(q)\)	\(=\)	\(q+1.81361 q^{2} +1.28917 q^{4} +2.81361 q^{5} -1.28917 q^{8} +O(q^{10})\)	\(q+1.81361 q^{2} +1.28917 q^{4} +2.81361 q^{5} -1.28917 q^{8} +5.10278 q^{10} -3.10278 q^{11} -1.00000 q^{13} -4.91638 q^{16} -0.524438 q^{17} -0.813607 q^{19} +3.62721 q^{20} -5.62721 q^{22} -7.33804 q^{23} +2.91638 q^{25} -1.81361 q^{26} -8.28917 q^{29} -1.39194 q^{31} -6.33804 q^{32} -0.951124 q^{34} -6.15165 q^{37} -1.47556 q^{38} -3.62721 q^{40} -4.20555 q^{41} +6.75971 q^{43} -4.00000 q^{44} -13.3083 q^{46} -5.97028 q^{47} +5.28917 q^{50} -1.28917 q^{52} +2.49472 q^{53} -8.72999 q^{55} -15.0333 q^{58} -4.47054 q^{59} +2.00000 q^{61} -2.52444 q^{62} -1.66196 q^{64} -2.81361 q^{65} +10.0383 q^{67} -0.676089 q^{68} +8.72999 q^{71} +2.34307 q^{73} -11.1567 q^{74} -1.04888 q^{76} -13.5436 q^{79} -13.8328 q^{80} -7.62721 q^{82} +16.4791 q^{83} -1.47556 q^{85} +12.2594 q^{86} +4.00000 q^{88} -10.6464 q^{89} -9.45998 q^{92} -10.8277 q^{94} -2.28917 q^{95} +1.18639 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + 3 * q^4 + 2 * q^5 - 3 * q^8	\( 3 q - q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{8} + 8 q^{10} - 2 q^{11} - 3 q^{13} - q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} - 4 q^{22} - 10 q^{23} - 5 q^{25} + q^{26} - 24 q^{29} + 4 q^{31} - 7 q^{32} - 14 q^{34} - 10 q^{38} + 2 q^{40} + 2 q^{41} + 10 q^{43} - 12 q^{44} - 18 q^{46} - 8 q^{47} + 15 q^{50} - 3 q^{52} - 8 q^{53} - 6 q^{55} + 12 q^{58} - 4 q^{59} + 6 q^{61} - 2 q^{62} - 17 q^{64} - 2 q^{65} - 12 q^{67} + 22 q^{68} + 6 q^{71} + 10 q^{73} - 30 q^{74} + 8 q^{76} - 14 q^{79} - 14 q^{80} - 10 q^{82} - 12 q^{83} - 10 q^{85} + 26 q^{86} + 12 q^{88} + 2 q^{89} + 12 q^{92} + 10 q^{94} - 6 q^{95} + 10 q^{97}+O(q^{100}) \) 3 * q - q^2 + 3 * q^4 + 2 * q^5 - 3 * q^8 + 8 * q^10 - 2 * q^11 - 3 * q^13 - q^16 + 4 * q^17 + 4 * q^19 - 2 * q^20 - 4 * q^22 - 10 * q^23 - 5 * q^25 + q^26 - 24 * q^29 + 4 * q^31 - 7 * q^32 - 14 * q^34 - 10 * q^38 + 2 * q^40 + 2 * q^41 + 10 * q^43 - 12 * q^44 - 18 * q^46 - 8 * q^47 + 15 * q^50 - 3 * q^52 - 8 * q^53 - 6 * q^55 + 12 * q^58 - 4 * q^59 + 6 * q^61 - 2 * q^62 - 17 * q^64 - 2 * q^65 - 12 * q^67 + 22 * q^68 + 6 * q^71 + 10 * q^73 - 30 * q^74 + 8 * q^76 - 14 * q^79 - 14 * q^80 - 10 * q^82 - 12 * q^83 - 10 * q^85 + 26 * q^86 + 12 * q^88 + 2 * q^89 + 12 * q^92 + 10 * q^94 - 6 * q^95 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more