LMFDB - Embedded newform 4851.2.a.bj.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4851,2,Mod(1,4851)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4851, 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("4851.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4851 = 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4851.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:	$38.7354300205$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	4851.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53209 q^{2} +0.347296 q^{4} -2.34730 q^{5} -2.53209 q^{8} -3.59627 q^{10} -1.00000 q^{11} -0.184793 q^{13} -4.57398 q^{16} +3.92127 q^{17} +0.773318 q^{19} -0.815207 q^{20} -1.53209 q^{22} -8.35504 q^{23} +0.509800 q^{25} -0.283119 q^{26} +8.17024 q^{29} +2.65270 q^{31} -1.94356 q^{32} +6.00774 q^{34} -6.82295 q^{37} +1.18479 q^{38} +5.94356 q^{40} -0.426022 q^{41} +1.18479 q^{43} -0.347296 q^{44} -12.8007 q^{46} +7.68004 q^{47} +0.781059 q^{50} -0.0641778 q^{52} -6.55438 q^{53} +2.34730 q^{55} +12.5175 q^{58} -0.204393 q^{59} +14.5817 q^{61} +4.06418 q^{62} +6.17024 q^{64} +0.433763 q^{65} +3.75877 q^{67} +1.36184 q^{68} +9.96585 q^{71} +0.120615 q^{73} -10.4534 q^{74} +0.268571 q^{76} +0.327696 q^{79} +10.7365 q^{80} -0.652704 q^{82} -3.35504 q^{83} -9.20439 q^{85} +1.81521 q^{86} +2.53209 q^{88} +4.65270 q^{89} -2.90167 q^{92} +11.7665 q^{94} -1.81521 q^{95} +13.1206 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{5} - 3 q^{8} + 3 q^{10} - 3 q^{11} + 3 q^{13} - 6 q^{16} + 3 q^{17} + 9 q^{19} - 6 q^{20} + 3 q^{25} - 9 q^{26} + 3 q^{29} + 9 q^{31} + 9 q^{32} - 6 q^{34} + 3 q^{40} - 9 q^{41} - 24 q^{46} + 3 q^{47}+ \cdots + 45 q^{97}+O(q^{100}) $ 3 * q - 6 * q^5 - 3 * q^8 + 3 * q^10 - 3 * q^11 + 3 * q^13 - 6 * q^16 + 3 * q^17 + 9 * q^19 - 6 * q^20 + 3 * q^25 - 9 * q^26 + 3 * q^29 + 9 * q^31 + 9 * q^32 - 6 * q^34 + 3 * q^40 - 9 * q^41 - 24 * q^46 + 3 * q^47 - 15 * q^50 + 9 * q^52 - 9 * q^53 + 6 * q^55 + 15 * q^58 + 12 * q^61 + 3 * q^62 - 3 * q^64 - 15 * q^65 + 21 * q^68 + 9 * q^71 + 6 * q^73 - 18 * q^74 - 9 * q^76 - 3 * q^79 + 27 * q^80 - 3 * q^82 + 15 * q^83 - 27 * q^85 + 9 * q^86 + 3 * q^88 + 15 * q^89 + 3 * q^92 - 9 * q^95 + 45 * 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 $

$2$	1.53209	1.08335	0.541675	−	0.840588i	$-0.317790\pi$
$2$	1.53209	1.08335	0.541675	+	0.840588i	$0.317790\pi$
$3$	0	0
$3$	0	0
$4$	0.347296	0.173648
$5$	−2.34730	−1.04974	−0.524871	−	0.851182i	$-0.675887\pi$
$5$	−2.34730	−1.04974	−0.524871	+	0.851182i	$0.675887\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.53209	−0.895229
$9$	0	0
$10$	−3.59627	−1.13724
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.184793	−0.0512522	−0.0256261	−	0.999672i	$-0.508158\pi$
$13$	−0.184793	−0.0512522	−0.0256261	+	0.999672i	$0.508158\pi$
$14$	0	0
$15$	0	0
$16$	−4.57398	−1.14349
$17$	3.92127	0.951049	0.475524	−	0.879703i	$-0.342258\pi$
$17$	3.92127	0.951049	0.475524	+	0.879703i	$0.342258\pi$
$18$	0	0
$19$	0.773318	0.177411	0.0887057	−	0.996058i	$-0.471727\pi$
$19$	0.773318	0.177411	0.0887057	+	0.996058i	$0.471727\pi$
$20$	−0.815207	−0.182286
$21$	0	0
$22$	−1.53209	−0.326642
$23$	−8.35504	−1.74215	−0.871073	−	0.491154i	$-0.836575\pi$
$23$	−8.35504	−1.74215	−0.871073	+	0.491154i	$0.836575\pi$
$24$	0	0
$25$	0.509800	0.101960
$26$	−0.283119	−0.0555241
$27$	0	0
$28$	0	0
$29$	8.17024	1.51718	0.758588	−	0.651570i	$-0.225889\pi$
$29$	8.17024	1.51718	0.758588	+	0.651570i	$0.225889\pi$
$30$	0	0
$31$	2.65270	0.476440	0.238220	−	0.971211i	$-0.423436\pi$
$31$	2.65270	0.476440	0.238220	+	0.971211i	$0.423436\pi$
$32$	−1.94356	−0.343577
$33$	0	0
$34$	6.00774	1.03032
$35$	0	0
$36$	0	0
$37$	−6.82295	−1.12169	−0.560843	−	0.827922i	$-0.689523\pi$
$37$	−6.82295	−1.12169	−0.560843	+	0.827922i	$0.689523\pi$
$38$	1.18479	0.192199
$39$	0	0
$40$	5.94356	0.939760
$41$	−0.426022	−0.0665335	−0.0332667	−	0.999447i	$-0.510591\pi$
$41$	−0.426022	−0.0665335	−0.0332667	+	0.999447i	$0.510591\pi$
$42$	0	0
$43$	1.18479	0.180679	0.0903396	−	0.995911i	$-0.471205\pi$
$43$	1.18479	0.180679	0.0903396	+	0.995911i	$0.471205\pi$
$44$	−0.347296	−0.0523569
$45$	0	0
$46$	−12.8007	−1.88735
$47$	7.68004	1.12025	0.560125	−	0.828408i	$-0.310753\pi$
$47$	7.68004	1.12025	0.560125	+	0.828408i	$0.310753\pi$
$48$	0	0
$49$	0	0
$50$	0.781059	0.110458
$51$	0	0
$52$	−0.0641778	−0.00889986
$53$	−6.55438	−0.900313	−0.450157	−	0.892950i	$-0.648632\pi$
$53$	−6.55438	−0.900313	−0.450157	+	0.892950i	$0.648632\pi$
$54$	0	0
$55$	2.34730	0.316509
$56$	0	0
$57$	0	0
$58$	12.5175	1.64363
$59$	−0.204393	−0.0266097	−0.0133048	−	0.999911i	$-0.504235\pi$
$59$	−0.204393	−0.0266097	−0.0133048	+	0.999911i	$0.504235\pi$
$60$	0	0
$61$	14.5817	1.86700	0.933499	−	0.358580i	$-0.116739\pi$
$61$	14.5817	1.86700	0.933499	+	0.358580i	$0.116739\pi$
$62$	4.06418	0.516151
$63$	0	0
$64$	6.17024	0.771281
$65$	0.433763	0.0538017
$66$	0	0
$67$	3.75877	0.459207	0.229603	−	0.973284i	$-0.426257\pi$
$67$	3.75877	0.459207	0.229603	+	0.973284i	$0.426257\pi$
$68$	1.36184	0.165148
$69$	0	0
$70$	0	0
$71$	9.96585	1.18273	0.591365	−	0.806404i	$-0.298589\pi$
$71$	9.96585	1.18273	0.591365	+	0.806404i	$0.298589\pi$
$72$	0	0
$73$	0.120615	0.0141169	0.00705844	−	0.999975i	$-0.497753\pi$
$73$	0.120615	0.0141169	0.00705844	+	0.999975i	$0.497753\pi$
$74$	−10.4534	−1.21518
$75$	0	0
$76$	0.268571	0.0308072
$77$	0	0
$78$	0	0
$79$	0.327696	0.0368687	0.0184343	−	0.999830i	$-0.494132\pi$
$79$	0.327696	0.0368687	0.0184343	+	0.999830i	$0.494132\pi$
$80$	10.7365	1.20038
$81$	0	0
$82$	−0.652704	−0.0720791
$83$	−3.35504	−0.368263	−0.184132	−	0.982902i	$-0.558947\pi$
$83$	−3.35504	−0.368263	−0.184132	+	0.982902i	$0.558947\pi$
$84$	0	0
$85$	−9.20439	−0.998357
$86$	1.81521	0.195739
$87$	0	0
$88$	2.53209	0.269922
$89$	4.65270	0.493186	0.246593	−	0.969119i	$-0.420689\pi$
$89$	4.65270	0.493186	0.246593	+	0.969119i	$0.420689\pi$
$90$	0	0
$91$	0	0
$92$	−2.90167	−0.302520
$93$	0	0
$94$	11.7665	1.21362
$95$	−1.81521	−0.186236
$96$	0	0
$97$	13.1206	1.33220	0.666098	−	0.745864i	$-0.267963\pi$
$97$	13.1206	1.33220	0.666098	+	0.745864i	$0.267963\pi$
$98$	0	0
$99$	0	0
$100$	0.177052	0.0177052
$101$	5.90167	0.587239	0.293619	−	0.955922i	$-0.405140\pi$
$101$	5.90167	0.587239	0.293619	+	0.955922i	$0.405140\pi$
$102$	0	0
$103$	0.921274	0.0907759	0.0453879	−	0.998969i	$-0.485548\pi$
$103$	0.921274	0.0907759	0.0453879	+	0.998969i	$0.485548\pi$
$104$	0.467911	0.0458825
$105$	0	0
$106$	−10.0419	−0.975354
$107$	10.3473	1.00031	0.500155	−	0.865936i	$-0.333276\pi$
$107$	10.3473	1.00031	0.500155	+	0.865936i	$0.333276\pi$
$108$	0	0
$109$	0.475652	0.0455592	0.0227796	−	0.999741i	$-0.492748\pi$
$109$	0.475652	0.0455592	0.0227796	+	0.999741i	$0.492748\pi$
$110$	3.59627	0.342891
$111$	0	0
$112$	0	0
$113$	3.14290	0.295659	0.147830	−	0.989013i	$-0.452771\pi$
$113$	3.14290	0.295659	0.147830	+	0.989013i	$0.452771\pi$
$114$	0	0
$115$	19.6117	1.82880
$116$	2.83750	0.263455
$117$	0	0
$118$	−0.313148	−0.0288276
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	22.3405	2.02261
$123$	0	0
$124$	0.921274	0.0827329
$125$	10.5398	0.942711
$126$	0	0
$127$	−0.445622	−0.0395426	−0.0197713	−	0.999805i	$-0.506294\pi$
$127$	−0.445622	−0.0395426	−0.0197713	+	0.999805i	$0.506294\pi$
$128$	13.3405	1.17914
$129$	0	0
$130$	0.664563	0.0582860
$131$	12.0865	1.05600	0.528000	−	0.849245i	$-0.322942\pi$
$131$	12.0865	1.05600	0.528000	+	0.849245i	$0.322942\pi$
$132$	0	0
$133$	0	0
$134$	5.75877	0.497482
$135$	0	0
$136$	−9.92902	−0.851406
$137$	16.3482	1.39672	0.698362	−	0.715745i	$-0.253913\pi$
$137$	16.3482	1.39672	0.698362	+	0.715745i	$0.253913\pi$
$138$	0	0
$139$	−20.7297	−1.75827	−0.879134	−	0.476575i	$-0.841878\pi$
$139$	−20.7297	−1.75827	−0.879134	+	0.476575i	$0.841878\pi$
$140$	0	0
$141$	0	0
$142$	15.2686	1.28131
$143$	0.184793	0.0154531
$144$	0	0
$145$	−19.1780	−1.59264
$146$	0.184793	0.0152935
$147$	0	0
$148$	−2.36959	−0.194779
$149$	0.714193	0.0585090	0.0292545	−	0.999572i	$-0.490687\pi$
$149$	0.714193	0.0585090	0.0292545	+	0.999572i	$0.490687\pi$
$150$	0	0
$151$	−9.54488	−0.776751	−0.388376	−	0.921501i	$-0.626964\pi$
$151$	−9.54488	−0.776751	−0.388376	+	0.921501i	$0.626964\pi$
$152$	−1.95811	−0.158824
$153$	0	0
$154$	0	0
$155$	−6.22668	−0.500139
$156$	0	0
$157$	9.24123	0.737530	0.368765	−	0.929523i	$-0.379781\pi$
$157$	9.24123	0.737530	0.368765	+	0.929523i	$0.379781\pi$
$158$	0.502059	0.0399417
$159$	0	0
$160$	4.56212	0.360667
$161$	0	0
$162$	0	0
$163$	−8.80335	−0.689531	−0.344766	−	0.938689i	$-0.612042\pi$
$163$	−8.80335	−0.689531	−0.344766	+	0.938689i	$0.612042\pi$
$164$	−0.147956	−0.0115534
$165$	0	0
$166$	−5.14022	−0.398958
$167$	−8.38238	−0.648648	−0.324324	−	0.945946i	$-0.605137\pi$
$167$	−8.38238	−0.648648	−0.324324	+	0.945946i	$0.605137\pi$
$168$	0	0
$169$	−12.9659	−0.997373
$170$	−14.1019	−1.08157
$171$	0	0
$172$	0.411474	0.0313746
$173$	−5.44562	−0.414023	−0.207012	−	0.978339i	$-0.566374\pi$
$173$	−5.44562	−0.414023	−0.207012	+	0.978339i	$0.566374\pi$
$174$	0	0
$175$	0	0
$176$	4.57398	0.344777
$177$	0	0
$178$	7.12836	0.534293
$179$	10.0915	0.754276	0.377138	−	0.926157i	$-0.376908\pi$
$179$	10.0915	0.754276	0.377138	+	0.926157i	$0.376908\pi$
$180$	0	0
$181$	19.8794	1.47762	0.738812	−	0.673912i	$-0.235387\pi$
$181$	19.8794	1.47762	0.738812	+	0.673912i	$0.235387\pi$
$182$	0	0
$183$	0	0
$184$	21.1557	1.55962
$185$	16.0155	1.17748
$186$	0	0
$187$	−3.92127	−0.286752
$188$	2.66725	0.194529
$189$	0	0
$190$	−2.78106	−0.201759
$191$	25.3851	1.83680	0.918399	−	0.395654i	$-0.129482\pi$
$191$	25.3851	1.83680	0.918399	+	0.395654i	$0.129482\pi$
$192$	0	0
$193$	−7.73143	−0.556520	−0.278260	−	0.960506i	$-0.589758\pi$
$193$	−7.73143	−0.556520	−0.278260	+	0.960506i	$0.589758\pi$
$194$	20.1019	1.44324
$195$	0	0
$196$	0	0
$197$	5.37464	0.382927	0.191464	−	0.981500i	$-0.438677\pi$
$197$	5.37464	0.382927	0.191464	+	0.981500i	$0.438677\pi$
$198$	0	0
$199$	8.78106	0.622473	0.311236	−	0.950333i	$-0.399257\pi$
$199$	8.78106	0.622473	0.311236	+	0.950333i	$0.399257\pi$
$200$	−1.29086	−0.0912775
$201$	0	0
$202$	9.04189	0.636185
$203$	0	0
$204$	0	0
$205$	1.00000	0.0698430
$206$	1.41147	0.0983421
$207$	0	0
$208$	0.845237	0.0586066
$209$	−0.773318	−0.0534916
$210$	0	0
$211$	4.24123	0.291978	0.145989	−	0.989286i	$-0.453364\pi$
$211$	4.24123	0.291978	0.145989	+	0.989286i	$0.453364\pi$
$212$	−2.27631	−0.156338
$213$	0	0
$214$	15.8530	1.08369
$215$	−2.78106	−0.189667
$216$	0	0
$217$	0	0
$218$	0.728741	0.0493566
$219$	0	0
$220$	0.815207	0.0549613
$221$	−0.724622	−0.0487434
$222$	0	0
$223$	−18.0351	−1.20772	−0.603859	−	0.797091i	$-0.706371\pi$
$223$	−18.0351	−1.20772	−0.603859	+	0.797091i	$0.706371\pi$
$224$	0	0
$225$	0	0
$226$	4.81521	0.320303
$227$	−18.2053	−1.20833	−0.604165	−	0.796859i	$-0.706493\pi$
$227$	−18.2053	−1.20833	−0.604165	+	0.796859i	$0.706493\pi$
$228$	0	0
$229$	19.2412	1.27150	0.635748	−	0.771897i	$-0.280692\pi$
$229$	19.2412	1.27150	0.635748	+	0.771897i	$0.280692\pi$
$230$	30.0469	1.98124
$231$	0	0
$232$	−20.6878	−1.35822
$233$	−9.32501	−0.610902	−0.305451	−	0.952208i	$-0.598807\pi$
$233$	−9.32501	−0.610902	−0.305451	+	0.952208i	$0.598807\pi$
$234$	0	0
$235$	−18.0273	−1.17597
$236$	−0.0709849	−0.00462072
$237$	0	0
$238$	0	0
$239$	−11.1898	−0.723811	−0.361905	−	0.932215i	$-0.617874\pi$
$239$	−11.1898	−0.723811	−0.361905	+	0.932215i	$0.617874\pi$
$240$	0	0
$241$	20.2044	1.30148	0.650740	−	0.759301i	$-0.274459\pi$
$241$	20.2044	1.30148	0.650740	+	0.759301i	$0.274459\pi$
$242$	1.53209	0.0984864
$243$	0	0
$244$	5.06418	0.324201
$245$	0	0
$246$	0	0
$247$	−0.142903	−0.00909273
$248$	−6.71688	−0.426522
$249$	0	0
$250$	16.1480	1.02129
$251$	5.89218	0.371911	0.185956	−	0.982558i	$-0.440462\pi$
$251$	5.89218	0.371911	0.185956	+	0.982558i	$0.440462\pi$
$252$	0	0
$253$	8.35504	0.525277
$254$	−0.682733	−0.0428385
$255$	0	0
$256$	8.09833	0.506145
$257$	2.73648	0.170697	0.0853485	−	0.996351i	$-0.472800\pi$
$257$	2.73648	0.170697	0.0853485	+	0.996351i	$0.472800\pi$
$258$	0	0
$259$	0	0
$260$	0.150644	0.00934256
$261$	0	0
$262$	18.5175	1.14402
$263$	−6.19759	−0.382160	−0.191080	−	0.981575i	$-0.561199\pi$
$263$	−6.19759	−0.382160	−0.191080	+	0.981575i	$0.561199\pi$
$264$	0	0
$265$	15.3851	0.945097
$266$	0	0
$267$	0	0
$268$	1.30541	0.0797404
$269$	4.20945	0.256654	0.128327	−	0.991732i	$-0.459039\pi$
$269$	4.20945	0.256654	0.128327	+	0.991732i	$0.459039\pi$
$270$	0	0
$271$	25.5817	1.55398	0.776989	−	0.629514i	$-0.216746\pi$
$271$	25.5817	1.55398	0.776989	+	0.629514i	$0.216746\pi$
$272$	−17.9358	−1.08752
$273$	0	0
$274$	25.0469	1.51314
$275$	−0.509800	−0.0307421
$276$	0	0
$277$	−15.3577	−0.922756	−0.461378	−	0.887204i	$-0.652645\pi$
$277$	−15.3577	−0.922756	−0.461378	+	0.887204i	$0.652645\pi$
$278$	−31.7597	−1.90482
$279$	0	0
$280$	0	0
$281$	−19.2540	−1.14860	−0.574299	−	0.818645i	$-0.694725\pi$
$281$	−19.2540	−1.14860	−0.574299	+	0.818645i	$0.694725\pi$
$282$	0	0
$283$	−27.0651	−1.60885	−0.804427	−	0.594052i	$-0.797527\pi$
$283$	−27.0651	−1.60885	−0.804427	+	0.594052i	$0.797527\pi$
$284$	3.46110	0.205379
$285$	0	0
$286$	0.283119	0.0167412
$287$	0	0
$288$	0	0
$289$	−1.62361	−0.0955063
$290$	−29.3824	−1.72539
$291$	0	0
$292$	0.0418891	0.00245137
$293$	−4.03508	−0.235732	−0.117866	−	0.993030i	$-0.537605\pi$
$293$	−4.03508	−0.235732	−0.117866	+	0.993030i	$0.537605\pi$
$294$	0	0
$295$	0.479771	0.0279333
$296$	17.2763	1.00417
$297$	0	0
$298$	1.09421	0.0633857
$299$	1.54395	0.0892888
$300$	0	0
$301$	0	0
$302$	−14.6236	−0.841494
$303$	0	0
$304$	−3.53714	−0.202869
$305$	−34.2276	−1.95987
$306$	0	0
$307$	2.92902	0.167168	0.0835839	−	0.996501i	$-0.473363\pi$
$307$	2.92902	0.167168	0.0835839	+	0.996501i	$0.473363\pi$
$308$	0	0
$309$	0	0
$310$	−9.53983	−0.541826
$311$	−15.6236	−0.885934	−0.442967	−	0.896538i	$-0.646074\pi$
$311$	−15.6236	−0.885934	−0.442967	+	0.896538i	$0.646074\pi$
$312$	0	0
$313$	−21.9760	−1.24215	−0.621077	−	0.783749i	$-0.713305\pi$
$313$	−21.9760	−1.24215	−0.621077	+	0.783749i	$0.713305\pi$
$314$	14.1584	0.799004
$315$	0	0
$316$	0.113808	0.00640218
$317$	3.01186	0.169163	0.0845814	−	0.996417i	$-0.473045\pi$
$317$	3.01186	0.169163	0.0845814	+	0.996417i	$0.473045\pi$
$318$	0	0
$319$	−8.17024	−0.457446
$320$	−14.4834	−0.809646
$321$	0	0
$322$	0	0
$323$	3.03239	0.168727
$324$	0	0
$325$	−0.0942073	−0.00522568
$326$	−13.4875	−0.747004
$327$	0	0
$328$	1.07873	0.0595627
$329$	0	0
$330$	0	0
$331$	27.4688	1.50982	0.754912	−	0.655826i	$-0.227679\pi$
$331$	27.4688	1.50982	0.754912	+	0.655826i	$0.227679\pi$
$332$	−1.16519	−0.0639482
$333$	0	0
$334$	−12.8425	−0.702713
$335$	−8.82295	−0.482049
$336$	0	0
$337$	17.2422	0.939240	0.469620	−	0.882869i	$-0.344391\pi$
$337$	17.2422	0.939240	0.469620	+	0.882869i	$0.344391\pi$
$338$	−19.8648	−1.08050
$339$	0	0
$340$	−3.19665	−0.173363
$341$	−2.65270	−0.143652
$342$	0	0
$343$	0	0
$344$	−3.00000	−0.161749
$345$	0	0
$346$	−8.34318	−0.448532
$347$	11.4311	0.613652	0.306826	−	0.951766i	$-0.400733\pi$
$347$	11.4311	0.613652	0.306826	+	0.951766i	$0.400733\pi$
$348$	0	0
$349$	28.3209	1.51598	0.757991	−	0.652265i	$-0.226181\pi$
$349$	28.3209	1.51598	0.757991	+	0.652265i	$0.226181\pi$
$350$	0	0
$351$	0	0
$352$	1.94356	0.103592
$353$	−16.4361	−0.874807	−0.437403	−	0.899265i	$-0.644102\pi$
$353$	−16.4361	−0.874807	−0.437403	+	0.899265i	$0.644102\pi$
$354$	0	0
$355$	−23.3928	−1.24156
$356$	1.61587	0.0856408
$357$	0	0
$358$	15.4611	0.817145
$359$	−24.8631	−1.31222	−0.656112	−	0.754664i	$-0.727800\pi$
$359$	−24.8631	−1.31222	−0.656112	+	0.754664i	$0.727800\pi$
$360$	0	0
$361$	−18.4020	−0.968525
$362$	30.4570	1.60078
$363$	0	0
$364$	0	0
$365$	−0.283119	−0.0148191
$366$	0	0
$367$	4.29860	0.224385	0.112193	−	0.993686i	$-0.464213\pi$
$367$	4.29860	0.224385	0.112193	+	0.993686i	$0.464213\pi$
$368$	38.2158	1.99213
$369$	0	0
$370$	24.5371	1.27563
$371$	0	0
$372$	0	0
$373$	−2.72967	−0.141337	−0.0706686	−	0.997500i	$-0.522513\pi$
$373$	−2.72967	−0.141337	−0.0706686	+	0.997500i	$0.522513\pi$
$374$	−6.00774	−0.310653
$375$	0	0
$376$	−19.4466	−1.00288
$377$	−1.50980	−0.0777587
$378$	0	0
$379$	−16.4584	−0.845412	−0.422706	−	0.906267i	$-0.638920\pi$
$379$	−16.4584	−0.845412	−0.422706	+	0.906267i	$0.638920\pi$
$380$	−0.630415	−0.0323396
$381$	0	0
$382$	38.8922	1.98990
$383$	7.85978	0.401616	0.200808	−	0.979631i	$-0.435643\pi$
$383$	7.85978	0.401616	0.200808	+	0.979631i	$0.435643\pi$
$384$	0	0
$385$	0	0
$386$	−11.8452	−0.602907
$387$	0	0
$388$	4.55674	0.231334
$389$	−9.64321	−0.488930	−0.244465	−	0.969658i	$-0.578612\pi$
$389$	−9.64321	−0.488930	−0.244465	+	0.969658i	$0.578612\pi$
$390$	0	0
$391$	−32.7624	−1.65687
$392$	0	0
$393$	0	0
$394$	8.23442	0.414844
$395$	−0.769200	−0.0387026
$396$	0	0
$397$	−9.43376	−0.473467	−0.236733	−	0.971575i	$-0.576077\pi$
$397$	−9.43376	−0.473467	−0.236733	+	0.971575i	$0.576077\pi$
$398$	13.4534	0.674356
$399$	0	0
$400$	−2.33181	−0.116591
$401$	18.2199	0.909857	0.454929	−	0.890528i	$-0.349665\pi$
$401$	18.2199	0.909857	0.454929	+	0.890528i	$0.349665\pi$
$402$	0	0
$403$	−0.490200	−0.0244186
$404$	2.04963	0.101973
$405$	0	0
$406$	0	0
$407$	6.82295	0.338201
$408$	0	0
$409$	6.41147	0.317027	0.158513	−	0.987357i	$-0.449330\pi$
$409$	6.41147	0.317027	0.158513	+	0.987357i	$0.449330\pi$
$410$	1.53209	0.0756645
$411$	0	0
$412$	0.319955	0.0157631
$413$	0	0
$414$	0	0
$415$	7.87527	0.386582
$416$	0.359156	0.0176091
$417$	0	0
$418$	−1.18479	−0.0579501
$419$	−1.62092	−0.0791871	−0.0395935	−	0.999216i	$-0.512606\pi$
$419$	−1.62092	−0.0791871	−0.0395935	+	0.999216i	$0.512606\pi$
$420$	0	0
$421$	31.1489	1.51810	0.759052	−	0.651030i	$-0.225663\pi$
$421$	31.1489	1.51810	0.759052	+	0.651030i	$0.225663\pi$
$422$	6.49794	0.316315
$423$	0	0
$424$	16.5963	0.805986
$425$	1.99907	0.0969690
$426$	0	0
$427$	0	0
$428$	3.59358	0.173702
$429$	0	0
$430$	−4.26083	−0.205475
$431$	−34.2080	−1.64774	−0.823871	−	0.566777i	$-0.808190\pi$
$431$	−34.2080	−1.64774	−0.823871	+	0.566777i	$0.808190\pi$
$432$	0	0
$433$	−20.4979	−0.985068	−0.492534	−	0.870293i	$-0.663929\pi$
$433$	−20.4979	−0.985068	−0.492534	+	0.870293i	$0.663929\pi$
$434$	0	0
$435$	0	0
$436$	0.165192	0.00791127
$437$	−6.46110	−0.309077
$438$	0	0
$439$	31.3218	1.49491	0.747455	−	0.664313i	$-0.231276\pi$
$439$	31.3218	1.49491	0.747455	+	0.664313i	$0.231276\pi$
$440$	−5.94356	−0.283348
$441$	0	0
$442$	−1.11019	−0.0528061
$443$	29.3979	1.39673	0.698367	−	0.715740i	$-0.253910\pi$
$443$	29.3979	1.39673	0.698367	+	0.715740i	$0.253910\pi$
$444$	0	0
$445$	−10.9213	−0.517718
$446$	−27.6313	−1.30838
$447$	0	0
$448$	0	0
$449$	6.90074	0.325666	0.162833	−	0.986654i	$-0.447937\pi$
$449$	6.90074	0.325666	0.162833	+	0.986654i	$0.447937\pi$
$450$	0	0
$451$	0.426022	0.0200606
$452$	1.09152	0.0513407
$453$	0	0
$454$	−27.8922	−1.30904
$455$	0	0
$456$	0	0
$457$	12.3746	0.578861	0.289431	−	0.957199i	$-0.406534\pi$
$457$	12.3746	0.578861	0.289431	+	0.957199i	$0.406534\pi$
$458$	29.4793	1.37748
$459$	0	0
$460$	6.81109	0.317569
$461$	−20.0942	−0.935881	−0.467940	−	0.883760i	$-0.655004\pi$
$461$	−20.0942	−0.935881	−0.467940	+	0.883760i	$0.655004\pi$
$462$	0	0
$463$	8.69190	0.403947	0.201974	−	0.979391i	$-0.435265\pi$
$463$	8.69190	0.403947	0.201974	+	0.979391i	$0.435265\pi$
$464$	−37.3705	−1.73488
$465$	0	0
$466$	−14.2867	−0.661820
$467$	−4.76920	−0.220692	−0.110346	−	0.993893i	$-0.535196\pi$
$467$	−4.76920	−0.220692	−0.110346	+	0.993893i	$0.535196\pi$
$468$	0	0
$469$	0	0
$470$	−27.6195	−1.27399
$471$	0	0
$472$	0.517541	0.0238218
$473$	−1.18479	−0.0544768
$474$	0	0
$475$	0.394238	0.0180889
$476$	0	0
$477$	0	0
$478$	−17.1438	−0.784141
$479$	39.6141	1.81001	0.905007	−	0.425396i	$-0.139865\pi$
$479$	39.6141	1.81001	0.905007	+	0.425396i	$0.139865\pi$
$480$	0	0
$481$	1.26083	0.0574889
$482$	30.9549	1.40996
$483$	0	0
$484$	0.347296	0.0157862
$485$	−30.7980	−1.39846
$486$	0	0
$487$	5.15570	0.233627	0.116813	−	0.993154i	$-0.462732\pi$
$487$	5.15570	0.233627	0.116813	+	0.993154i	$0.462732\pi$
$488$	−36.9222	−1.67139
$489$	0	0
$490$	0	0
$491$	−10.1925	−0.459983	−0.229991	−	0.973193i	$-0.573870\pi$
$491$	−10.1925	−0.459983	−0.229991	+	0.973193i	$0.573870\pi$
$492$	0	0
$493$	32.0378	1.44291
$494$	−0.218941	−0.00985061
$495$	0	0
$496$	−12.1334	−0.544806
$497$	0	0
$498$	0	0
$499$	5.88981	0.263664	0.131832	−	0.991272i	$-0.457914\pi$
$499$	5.88981	0.263664	0.131832	+	0.991272i	$0.457914\pi$
$500$	3.66044	0.163700
$501$	0	0
$502$	9.02734	0.402910
$503$	12.8348	0.572276	0.286138	−	0.958188i	$-0.407628\pi$
$503$	12.8348	0.572276	0.286138	+	0.958188i	$0.407628\pi$
$504$	0	0
$505$	−13.8530	−0.616449
$506$	12.8007	0.569059
$507$	0	0
$508$	−0.154763	−0.00686650
$509$	−32.2131	−1.42782	−0.713910	−	0.700238i	$-0.753077\pi$
$509$	−32.2131	−1.42782	−0.713910	+	0.700238i	$0.753077\pi$
$510$	0	0
$511$	0	0
$512$	−14.2736	−0.630811
$513$	0	0
$514$	4.19253	0.184925
$515$	−2.16250	−0.0952913
$516$	0	0
$517$	−7.68004	−0.337768
$518$	0	0
$519$	0	0
$520$	−1.09833	−0.0481648
$521$	−18.5280	−0.811725	−0.405863	−	0.913934i	$-0.633029\pi$
$521$	−18.5280	−0.811725	−0.405863	+	0.913934i	$0.633029\pi$
$522$	0	0
$523$	−37.2422	−1.62849	−0.814243	−	0.580524i	$-0.802848\pi$
$523$	−37.2422	−1.62849	−0.814243	+	0.580524i	$0.802848\pi$
$524$	4.19759	0.183372
$525$	0	0
$526$	−9.49525	−0.414013
$527$	10.4020	0.453117
$528$	0	0
$529$	46.8066	2.03507
$530$	23.5713	1.02387
$531$	0	0
$532$	0	0
$533$	0.0787257	0.00340999
$534$	0	0
$535$	−24.2882	−1.05007
$536$	−9.51754	−0.411095
$537$	0	0
$538$	6.44924	0.278047
$539$	0	0
$540$	0	0
$541$	36.1566	1.55449	0.777247	−	0.629195i	$-0.216615\pi$
$541$	36.1566	1.55449	0.777247	+	0.629195i	$0.216615\pi$
$542$	39.1935	1.68350
$543$	0	0
$544$	−7.62124	−0.326758
$545$	−1.11650	−0.0478254
$546$	0	0
$547$	12.7980	0.547202	0.273601	−	0.961843i	$-0.411785\pi$
$547$	12.7980	0.547202	0.273601	+	0.961843i	$0.411785\pi$
$548$	5.67768	0.242538
$549$	0	0
$550$	−0.781059	−0.0333045
$551$	6.31820	0.269164
$552$	0	0
$553$	0	0
$554$	−23.5294	−0.999668
$555$	0	0
$556$	−7.19934	−0.305320
$557$	30.6587	1.29905	0.649525	−	0.760340i	$-0.274968\pi$
$557$	30.6587	1.29905	0.649525	+	0.760340i	$0.274968\pi$
$558$	0	0
$559$	−0.218941	−0.00926021
$560$	0	0
$561$	0	0
$562$	−29.4989	−1.24433
$563$	14.5047	0.611302	0.305651	−	0.952144i	$-0.401126\pi$
$563$	14.5047	0.611302	0.305651	+	0.952144i	$0.401126\pi$
$564$	0	0
$565$	−7.37733	−0.310366
$566$	−41.4662	−1.74295
$567$	0	0
$568$	−25.2344	−1.05881
$569$	−19.0051	−0.796733	−0.398367	−	0.917226i	$-0.630423\pi$
$569$	−19.0051	−0.796733	−0.398367	+	0.917226i	$0.630423\pi$
$570$	0	0
$571$	−21.0009	−0.878862	−0.439431	−	0.898276i	$-0.644820\pi$
$571$	−21.0009	−0.878862	−0.439431	+	0.898276i	$0.644820\pi$
$572$	0.0641778	0.00268341
$573$	0	0
$574$	0	0
$575$	−4.25940	−0.177629
$576$	0	0
$577$	−5.18748	−0.215958	−0.107979	−	0.994153i	$-0.534438\pi$
$577$	−5.18748	−0.215958	−0.107979	+	0.994153i	$0.534438\pi$
$578$	−2.48751	−0.103467
$579$	0	0
$580$	−6.66044	−0.276560
$581$	0	0
$582$	0	0
$583$	6.55438	0.271455
$584$	−0.305407	−0.0126378
$585$	0	0
$586$	−6.18210	−0.255380
$587$	−8.33813	−0.344151	−0.172076	−	0.985084i	$-0.555047\pi$
$587$	−8.33813	−0.344151	−0.172076	+	0.985084i	$0.555047\pi$
$588$	0	0
$589$	2.05138	0.0845258
$590$	0.735051	0.0302616
$591$	0	0
$592$	31.2080	1.28264
$593$	−6.10195	−0.250577	−0.125288	−	0.992120i	$-0.539986\pi$
$593$	−6.10195	−0.250577	−0.125288	+	0.992120i	$0.539986\pi$
$594$	0	0
$595$	0	0
$596$	0.248037	0.0101600
$597$	0	0
$598$	2.36547	0.0967311
$599$	13.8503	0.565907	0.282954	−	0.959134i	$-0.408686\pi$
$599$	13.8503	0.565907	0.282954	+	0.959134i	$0.408686\pi$
$600$	0	0
$601$	31.2695	1.27551	0.637755	−	0.770239i	$-0.279863\pi$
$601$	31.2695	1.27551	0.637755	+	0.770239i	$0.279863\pi$
$602$	0	0
$603$	0	0
$604$	−3.31490	−0.134881
$605$	−2.34730	−0.0954312
$606$	0	0
$607$	37.7962	1.53410	0.767051	−	0.641587i	$-0.221724\pi$
$607$	37.7962	1.53410	0.767051	+	0.641587i	$0.221724\pi$
$608$	−1.50299	−0.0609544
$609$	0	0
$610$	−52.4397	−2.12322
$611$	−1.41921	−0.0574153
$612$	0	0
$613$	21.1343	0.853608	0.426804	−	0.904344i	$-0.359639\pi$
$613$	21.1343	0.853608	0.426804	+	0.904344i	$0.359639\pi$
$614$	4.48751	0.181101
$615$	0	0
$616$	0	0
$617$	−23.7743	−0.957115	−0.478558	−	0.878056i	$-0.658840\pi$
$617$	−23.7743	−0.957115	−0.478558	+	0.878056i	$0.658840\pi$
$618$	0	0
$619$	13.5021	0.542694	0.271347	−	0.962482i	$-0.412531\pi$
$619$	13.5021	0.542694	0.271347	+	0.962482i	$0.412531\pi$
$620$	−2.16250	−0.0868482
$621$	0	0
$622$	−23.9368	−0.959776
$623$	0	0
$624$	0	0
$625$	−27.2891	−1.09156
$626$	−33.6691	−1.34569
$627$	0	0
$628$	3.20945	0.128071
$629$	−26.7547	−1.06678
$630$	0	0
$631$	−38.2354	−1.52213	−0.761063	−	0.648678i	$-0.775322\pi$
$631$	−38.2354	−1.52213	−0.761063	+	0.648678i	$0.775322\pi$
$632$	−0.829755	−0.0330059
$633$	0	0
$634$	4.61444	0.183263
$635$	1.04601	0.0415096
$636$	0	0
$637$	0	0
$638$	−12.5175	−0.495574
$639$	0	0
$640$	−31.3141	−1.23780
$641$	−38.5553	−1.52284	−0.761422	−	0.648257i	$-0.775498\pi$
$641$	−38.5553	−1.52284	−0.761422	+	0.648257i	$0.775498\pi$
$642$	0	0
$643$	19.2627	0.759647	0.379823	−	0.925059i	$-0.375985\pi$
$643$	19.2627	0.759647	0.379823	+	0.925059i	$0.375985\pi$
$644$	0	0
$645$	0	0
$646$	4.64590	0.182790
$647$	−39.7844	−1.56408	−0.782042	−	0.623225i	$-0.785822\pi$
$647$	−39.7844	−1.56408	−0.782042	+	0.623225i	$0.785822\pi$
$648$	0	0
$649$	0.204393	0.00802312
$650$	−0.144334	−0.00566124
$651$	0	0
$652$	−3.05737	−0.119736
$653$	−4.76651	−0.186528	−0.0932640	−	0.995641i	$-0.529730\pi$
$653$	−4.76651	−0.186528	−0.0932640	+	0.995641i	$0.529730\pi$
$654$	0	0
$655$	−28.3705	−1.10853
$656$	1.94862	0.0760807
$657$	0	0
$658$	0	0
$659$	38.3550	1.49410	0.747050	−	0.664768i	$-0.231470\pi$
$659$	38.3550	1.49410	0.747050	+	0.664768i	$0.231470\pi$
$660$	0	0
$661$	−10.3628	−0.403065	−0.201533	−	0.979482i	$-0.564592\pi$
$661$	−10.3628	−0.403065	−0.201533	+	0.979482i	$0.564592\pi$
$662$	42.0847	1.63567
$663$	0	0
$664$	8.49525	0.329680
$665$	0	0
$666$	0	0
$667$	−68.2627	−2.64314
$668$	−2.91117	−0.112637
$669$	0	0
$670$	−13.5175	−0.522228
$671$	−14.5817	−0.562921
$672$	0	0
$673$	−25.6168	−0.987455	−0.493728	−	0.869617i	$-0.664366\pi$
$673$	−25.6168	−0.987455	−0.493728	+	0.869617i	$0.664366\pi$
$674$	26.4165	1.01753
$675$	0	0
$676$	−4.50299	−0.173192
$677$	−41.8375	−1.60795	−0.803973	−	0.594666i	$-0.797284\pi$
$677$	−41.8375	−1.60795	−0.803973	+	0.594666i	$0.797284\pi$
$678$	0	0
$679$	0	0
$680$	23.3063	0.893757
$681$	0	0
$682$	−4.06418	−0.155625
$683$	46.0428	1.76178	0.880890	−	0.473321i	$-0.156945\pi$
$683$	46.0428	1.76178	0.880890	+	0.473321i	$0.156945\pi$
$684$	0	0
$685$	−38.3741	−1.46620
$686$	0	0
$687$	0	0
$688$	−5.41921	−0.206606
$689$	1.21120	0.0461430
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	−1.89124	−0.0718943
$693$	0	0
$694$	17.5134	0.664800
$695$	48.6587	1.84573
$696$	0	0
$697$	−1.67055	−0.0632766
$698$	43.3901	1.64234
$699$	0	0
$700$	0	0
$701$	32.8144	1.23938	0.619691	−	0.784846i	$-0.287258\pi$
$701$	32.8144	1.23938	0.619691	+	0.784846i	$0.287258\pi$
$702$	0	0
$703$	−5.27631	−0.199000
$704$	−6.17024	−0.232550
$705$	0	0
$706$	−25.1816	−0.947722
$707$	0	0
$708$	0	0
$709$	7.40198	0.277987	0.138994	−	0.990293i	$-0.455613\pi$
$709$	7.40198	0.277987	0.138994	+	0.990293i	$0.455613\pi$
$710$	−35.8399	−1.34505
$711$	0	0
$712$	−11.7811	−0.441514
$713$	−22.1634	−0.830027
$714$	0	0
$715$	−0.433763	−0.0162218
$716$	3.50475	0.130979
$717$	0	0
$718$	−38.0925	−1.42160
$719$	14.1370	0.527222	0.263611	−	0.964629i	$-0.415086\pi$
$719$	14.1370	0.527222	0.263611	+	0.964629i	$0.415086\pi$
$720$	0	0
$721$	0	0
$722$	−28.1935	−1.04925
$723$	0	0
$724$	6.90404	0.256587
$725$	4.16519	0.154691
$726$	0	0
$727$	15.2790	0.566667	0.283333	−	0.959021i	$-0.408560\pi$
$727$	15.2790	0.566667	0.283333	+	0.959021i	$0.408560\pi$
$728$	0	0
$729$	0	0
$730$	−0.433763	−0.0160543
$731$	4.64590	0.171835
$732$	0	0
$733$	1.34493	0.0496762	0.0248381	−	0.999691i	$-0.492093\pi$
$733$	1.34493	0.0496762	0.0248381	+	0.999691i	$0.492093\pi$
$734$	6.58584	0.243088
$735$	0	0
$736$	16.2385	0.598561
$737$	−3.75877	−0.138456
$738$	0	0
$739$	−25.6536	−0.943684	−0.471842	−	0.881683i	$-0.656411\pi$
$739$	−25.6536	−0.943684	−0.471842	+	0.881683i	$0.656411\pi$
$740$	5.56212	0.204468
$741$	0	0
$742$	0	0
$743$	6.08109	0.223094	0.111547	−	0.993759i	$-0.464420\pi$
$743$	6.08109	0.223094	0.111547	+	0.993759i	$0.464420\pi$
$744$	0	0
$745$	−1.67642	−0.0614194
$746$	−4.18210	−0.153118
$747$	0	0
$748$	−1.36184	−0.0497940
$749$	0	0
$750$	0	0
$751$	20.8108	0.759396	0.379698	−	0.925111i	$-0.376028\pi$
$751$	20.8108	0.759396	0.379698	+	0.925111i	$0.376028\pi$
$752$	−35.1284	−1.28100
$753$	0	0
$754$	−2.31315	−0.0842399
$755$	22.4047	0.815389
$756$	0	0
$757$	−5.25402	−0.190961	−0.0954804	−	0.995431i	$-0.530439\pi$
$757$	−5.25402	−0.190961	−0.0954804	+	0.995431i	$0.530439\pi$
$758$	−25.2158	−0.915877
$759$	0	0
$760$	4.59627	0.166724
$761$	40.9813	1.48557	0.742786	−	0.669529i	$-0.233504\pi$
$761$	40.9813	1.48557	0.742786	+	0.669529i	$0.233504\pi$
$762$	0	0
$763$	0	0
$764$	8.81614	0.318957
$765$	0	0
$766$	12.0419	0.435091
$767$	0.0377703	0.00136381
$768$	0	0
$769$	47.5580	1.71499	0.857493	−	0.514496i	$-0.172021\pi$
$769$	47.5580	1.71499	0.857493	+	0.514496i	$0.172021\pi$
$770$	0	0
$771$	0	0
$772$	−2.68510	−0.0966388
$773$	−20.4929	−0.737078	−0.368539	−	0.929612i	$-0.620142\pi$
$773$	−20.4929	−0.737078	−0.368539	+	0.929612i	$0.620142\pi$
$774$	0	0
$775$	1.35235	0.0485778
$776$	−33.2226	−1.19262
$777$	0	0
$778$	−14.7743	−0.529683
$779$	−0.329451	−0.0118038
$780$	0	0
$781$	−9.96585	−0.356606
$782$	−50.1949	−1.79497
$783$	0	0
$784$	0	0
$785$	−21.6919	−0.774217
$786$	0	0
$787$	−19.1206	−0.681576	−0.340788	−	0.940140i	$-0.610694\pi$
$787$	−19.1206	−0.681576	−0.340788	+	0.940140i	$0.610694\pi$
$788$	1.86659	0.0664946
$789$	0	0
$790$	−1.17848	−0.0419285
$791$	0	0
$792$	0	0
$793$	−2.69459	−0.0956878
$794$	−14.4534	−0.512931
$795$	0	0
$796$	3.04963	0.108091
$797$	52.5144	1.86015	0.930077	−	0.367365i	$-0.119740\pi$
$797$	52.5144	1.86015	0.930077	+	0.367365i	$0.119740\pi$
$798$	0	0
$799$	30.1156	1.06541
$800$	−0.990829	−0.0350311
$801$	0	0
$802$	27.9145	0.985694
$803$	−0.120615	−0.00425640
$804$	0	0
$805$	0	0
$806$	−0.751030	−0.0264539
$807$	0	0
$808$	−14.9436	−0.525713
$809$	−40.6715	−1.42993	−0.714967	−	0.699159i	$-0.753558\pi$
$809$	−40.6715	−1.42993	−0.714967	+	0.699159i	$0.753558\pi$
$810$	0	0
$811$	−20.5648	−0.722128	−0.361064	−	0.932541i	$-0.617586\pi$
$811$	−20.5648	−0.722128	−0.361064	+	0.932541i	$0.617586\pi$
$812$	0	0
$813$	0	0
$814$	10.4534	0.366390
$815$	20.6641	0.723831
$816$	0	0
$817$	0.916222	0.0320545
$818$	9.82295	0.343451
$819$	0	0
$820$	0.347296	0.0121281
$821$	−6.36278	−0.222062	−0.111031	−	0.993817i	$-0.535415\pi$
$821$	−6.36278	−0.222062	−0.111031	+	0.993817i	$0.535415\pi$
$822$	0	0
$823$	−2.50887	−0.0874536	−0.0437268	−	0.999044i	$-0.513923\pi$
$823$	−2.50887	−0.0874536	−0.0437268	+	0.999044i	$0.513923\pi$
$824$	−2.33275	−0.0812651
$825$	0	0
$826$	0	0
$827$	41.2645	1.43491	0.717453	−	0.696607i	$-0.245308\pi$
$827$	41.2645	1.43491	0.717453	+	0.696607i	$0.245308\pi$
$828$	0	0
$829$	−46.7779	−1.62466	−0.812331	−	0.583196i	$-0.801802\pi$
$829$	−46.7779	−1.62466	−0.812331	+	0.583196i	$0.801802\pi$
$830$	12.0656	0.418803
$831$	0	0
$832$	−1.14022	−0.0395298
$833$	0	0
$834$	0	0
$835$	19.6759	0.680913
$836$	−0.268571	−0.00928871
$837$	0	0
$838$	−2.48339	−0.0857874
$839$	−16.5439	−0.571161	−0.285580	−	0.958355i	$-0.592186\pi$
$839$	−16.5439	−0.571161	−0.285580	+	0.958355i	$0.592186\pi$
$840$	0	0
$841$	37.7529	1.30182
$842$	47.7229	1.64464
$843$	0	0
$844$	1.47296	0.0507015
$845$	30.4347	1.04699
$846$	0	0
$847$	0	0
$848$	29.9796	1.02950
$849$	0	0
$850$	3.06275	0.105051
$851$	57.0060	1.95414
$852$	0	0
$853$	13.2163	0.452516	0.226258	−	0.974067i	$-0.427351\pi$
$853$	13.2163	0.452516	0.226258	+	0.974067i	$0.427351\pi$
$854$	0	0
$855$	0	0
$856$	−26.2003	−0.895507
$857$	50.7211	1.73260	0.866300	−	0.499524i	$-0.166492\pi$
$857$	50.7211	1.73260	0.866300	+	0.499524i	$0.166492\pi$
$858$	0	0
$859$	55.1124	1.88041	0.940205	−	0.340609i	$-0.110633\pi$
$859$	55.1124	1.88041	0.940205	+	0.340609i	$0.110633\pi$
$860$	−0.965852	−0.0329353
$861$	0	0
$862$	−52.4097	−1.78508
$863$	−10.5202	−0.358113	−0.179056	−	0.983839i	$-0.557304\pi$
$863$	−10.5202	−0.358113	−0.179056	+	0.983839i	$0.557304\pi$
$864$	0	0
$865$	12.7825	0.434618
$866$	−31.4047	−1.06717
$867$	0	0
$868$	0	0
$869$	−0.327696	−0.0111163
$870$	0	0
$871$	−0.694593	−0.0235354
$872$	−1.20439	−0.0407859
$873$	0	0
$874$	−9.89899	−0.334838
$875$	0	0
$876$	0	0
$877$	−39.6536	−1.33901	−0.669504	−	0.742808i	$-0.733493\pi$
$877$	−39.6536	−1.33901	−0.669504	+	0.742808i	$0.733493\pi$
$878$	47.9878	1.61951
$879$	0	0
$880$	−10.7365	−0.361927
$881$	−25.0077	−0.842532	−0.421266	−	0.906937i	$-0.638414\pi$
$881$	−25.0077	−0.842532	−0.421266	+	0.906937i	$0.638414\pi$
$882$	0	0
$883$	50.9341	1.71407	0.857034	−	0.515260i	$-0.172305\pi$
$883$	50.9341	1.71407	0.857034	+	0.515260i	$0.172305\pi$
$884$	−0.251659	−0.00846420
$885$	0	0
$886$	45.0401	1.51315
$887$	−32.0360	−1.07566	−0.537832	−	0.843052i	$-0.680757\pi$
$887$	−32.0360	−1.07566	−0.537832	+	0.843052i	$0.680757\pi$
$888$	0	0
$889$	0	0
$890$	−16.7324	−0.560870
$891$	0	0
$892$	−6.26352	−0.209718
$893$	5.93912	0.198745
$894$	0	0
$895$	−23.6878	−0.791795
$896$	0	0
$897$	0	0
$898$	10.5725	0.352810
$899$	21.6732	0.722843
$900$	0	0
$901$	−25.7015	−0.856242
$902$	0.652704	0.0217327
$903$	0	0
$904$	−7.95811	−0.264683
$905$	−46.6628	−1.55112
$906$	0	0
$907$	−16.4730	−0.546976	−0.273488	−	0.961875i	$-0.588177\pi$
$907$	−16.4730	−0.546976	−0.273488	+	0.961875i	$0.588177\pi$
$908$	−6.32264	−0.209824
$909$	0	0
$910$	0	0
$911$	45.3979	1.50410	0.752049	−	0.659107i	$-0.229066\pi$
$911$	45.3979	1.50410	0.752049	+	0.659107i	$0.229066\pi$
$912$	0	0
$913$	3.35504	0.111036
$914$	18.9590	0.627109
$915$	0	0
$916$	6.68241	0.220793
$917$	0	0
$918$	0	0
$919$	−12.3209	−0.406429	−0.203214	−	0.979134i	$-0.565139\pi$
$919$	−12.3209	−0.406429	−0.203214	+	0.979134i	$0.565139\pi$
$920$	−49.6587	−1.63720
$921$	0	0
$922$	−30.7861	−1.01389
$923$	−1.84161	−0.0606175
$924$	0	0
$925$	−3.47834	−0.114367
$926$	13.3168	0.437616
$927$	0	0
$928$	−15.8794	−0.521266
$929$	−5.93313	−0.194660	−0.0973299	−	0.995252i	$-0.531030\pi$
$929$	−5.93313	−0.194660	−0.0973299	+	0.995252i	$0.531030\pi$
$930$	0	0
$931$	0	0
$932$	−3.23854	−0.106082
$933$	0	0
$934$	−7.30684	−0.239087
$935$	9.20439	0.301016
$936$	0	0
$937$	−45.0242	−1.47088	−0.735438	−	0.677593i	$-0.763023\pi$
$937$	−45.0242	−1.47088	−0.735438	+	0.677593i	$0.763023\pi$
$938$	0	0
$939$	0	0
$940$	−6.26083	−0.204206
$941$	6.06242	0.197629	0.0988147	−	0.995106i	$-0.468495\pi$
$941$	6.06242	0.197629	0.0988147	+	0.995106i	$0.468495\pi$
$942$	0	0
$943$	3.55943	0.115911
$944$	0.934889	0.0304280
$945$	0	0
$946$	−1.81521	−0.0590175
$947$	18.9426	0.615553	0.307776	−	0.951459i	$-0.400415\pi$
$947$	18.9426	0.615553	0.307776	+	0.951459i	$0.400415\pi$
$948$	0	0
$949$	−0.0222887	−0.000723522 0
$950$	0.604007	0.0195966
$951$	0	0
$952$	0	0
$953$	12.4970	0.404818	0.202409	−	0.979301i	$-0.435123\pi$
$953$	12.4970	0.404818	0.202409	+	0.979301i	$0.435123\pi$
$954$	0	0
$955$	−59.5863	−1.92817
$956$	−3.88619	−0.125688
$957$	0	0
$958$	60.6923	1.96088
$959$	0	0
$960$	0	0
$961$	−23.9632	−0.773005
$962$	1.93170	0.0622806
$963$	0	0
$964$	7.01691	0.226000
$965$	18.1480	0.584203
$966$	0	0
$967$	−18.5294	−0.595865	−0.297933	−	0.954587i	$-0.596297\pi$
$967$	−18.5294	−0.595865	−0.297933	+	0.954587i	$0.596297\pi$
$968$	−2.53209	−0.0813844
$969$	0	0
$970$	−47.1852	−1.51503
$971$	−41.4739	−1.33096	−0.665480	−	0.746415i	$-0.731773\pi$
$971$	−41.4739	−1.33096	−0.665480	+	0.746415i	$0.731773\pi$
$972$	0	0
$973$	0	0
$974$	7.89899	0.253100
$975$	0	0
$976$	−66.6965	−2.13490
$977$	22.2321	0.711267	0.355633	−	0.934626i	$-0.384265\pi$
$977$	22.2321	0.711267	0.355633	+	0.934626i	$0.384265\pi$
$978$	0	0
$979$	−4.65270	−0.148701
$980$	0	0
$981$	0	0
$982$	−15.6159	−0.498322
$983$	−35.4124	−1.12948	−0.564740	−	0.825269i	$-0.691024\pi$
$983$	−35.4124	−1.12948	−0.564740	+	0.825269i	$0.691024\pi$
$984$	0	0
$985$	−12.6159	−0.401975
$986$	49.0847	1.56318
$987$	0	0
$988$	−0.0496299	−0.00157894
$989$	−9.89899	−0.314769
$990$	0	0
$991$	45.5725	1.44766	0.723830	−	0.689979i	$-0.242380\pi$
$991$	45.5725	1.44766	0.723830	+	0.689979i	$0.242380\pi$
$992$	−5.15570	−0.163694
$993$	0	0
$994$	0	0
$995$	−20.6117	−0.653436
$996$	0	0
$997$	22.8367	0.723245	0.361622	−	0.932325i	$-0.382223\pi$
$997$	22.8367	0.723245	0.361622	+	0.932325i	$0.382223\pi$
$998$	9.02372	0.285641
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.bj.1.3		3
3.2	odd	2		539.2.a.j.1.1		3
7.2	even	3		693.2.i.h.298.1		6
7.4	even	3		693.2.i.h.100.1		6
7.6	odd	2		4851.2.a.bk.1.3		3
12.11	even	2		8624.2.a.ch.1.1		3
21.2	odd	6		77.2.e.a.67.3	yes	6
21.5	even	6		539.2.e.m.67.3		6
21.11	odd	6		77.2.e.a.23.3	✓	6
21.17	even	6		539.2.e.m.177.3		6
21.20	even	2		539.2.a.g.1.1		3
33.32	even	2		5929.2.a.x.1.3		3
84.11	even	6		1232.2.q.m.177.3		6
84.23	even	6		1232.2.q.m.529.3		6
84.83	odd	2		8624.2.a.co.1.3		3
231.2	even	30		847.2.n.f.81.1		24
231.32	even	6		847.2.e.c.485.1		6
231.53	odd	30		847.2.n.g.807.1		24
231.65	even	6		847.2.e.c.606.1		6
231.74	even	30		847.2.n.f.9.3		24
231.86	odd	30		847.2.n.g.81.3		24
231.95	even	30		847.2.n.f.632.1		24
231.107	even	30		847.2.n.f.130.1		24
231.116	even	30		847.2.n.f.366.1		24
231.128	even	30		847.2.n.f.753.3		24
231.137	odd	30		847.2.n.g.366.3		24
231.149	even	30		847.2.n.f.487.3		24
231.158	odd	30		847.2.n.g.632.3		24
231.170	odd	30		847.2.n.g.487.1		24
231.179	odd	30		847.2.n.g.9.1		24
231.191	odd	30		847.2.n.g.753.1		24
231.200	even	30		847.2.n.f.807.3		24
231.212	odd	30		847.2.n.g.130.3		24
231.230	odd	2		5929.2.a.u.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.e.a.23.3	✓	6	21.11	odd	6
77.2.e.a.67.3	yes	6	21.2	odd	6
539.2.a.g.1.1		3	21.20	even	2
539.2.a.j.1.1		3	3.2	odd	2
539.2.e.m.67.3		6	21.5	even	6
539.2.e.m.177.3		6	21.17	even	6
693.2.i.h.100.1		6	7.4	even	3
693.2.i.h.298.1		6	7.2	even	3
847.2.e.c.485.1		6	231.32	even	6
847.2.e.c.606.1		6	231.65	even	6
847.2.n.f.9.3		24	231.74	even	30
847.2.n.f.81.1		24	231.2	even	30
847.2.n.f.130.1		24	231.107	even	30
847.2.n.f.366.1		24	231.116	even	30
847.2.n.f.487.3		24	231.149	even	30
847.2.n.f.632.1		24	231.95	even	30
847.2.n.f.753.3		24	231.128	even	30
847.2.n.f.807.3		24	231.200	even	30
847.2.n.g.9.1		24	231.179	odd	30
847.2.n.g.81.3		24	231.86	odd	30
847.2.n.g.130.3		24	231.212	odd	30
847.2.n.g.366.3		24	231.137	odd	30
847.2.n.g.487.1		24	231.170	odd	30
847.2.n.g.632.3		24	231.158	odd	30
847.2.n.g.753.1		24	231.191	odd	30
847.2.n.g.807.1		24	231.53	odd	30
1232.2.q.m.177.3		6	84.11	even	6
1232.2.q.m.529.3		6	84.23	even	6
4851.2.a.bj.1.3		3	1.1	even	1	trivial
4851.2.a.bk.1.3		3	7.6	odd	2
5929.2.a.u.1.3		3	231.230	odd	2
5929.2.a.x.1.3		3	33.32	even	2
8624.2.a.ch.1.1		3	12.11	even	2
8624.2.a.co.1.3		3	84.83	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:	\( N \)	\(=\)	\( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4851.a (trivial)

\(f(q)\)	\(=\)	\(q+1.53209 q^{2} +0.347296 q^{4} -2.34730 q^{5} -2.53209 q^{8} -3.59627 q^{10} -1.00000 q^{11} -0.184793 q^{13} -4.57398 q^{16} +3.92127 q^{17} +0.773318 q^{19} -0.815207 q^{20} -1.53209 q^{22} -8.35504 q^{23} +0.509800 q^{25} -0.283119 q^{26} +8.17024 q^{29} +2.65270 q^{31} -1.94356 q^{32} +6.00774 q^{34} -6.82295 q^{37} +1.18479 q^{38} +5.94356 q^{40} -0.426022 q^{41} +1.18479 q^{43} -0.347296 q^{44} -12.8007 q^{46} +7.68004 q^{47} +0.781059 q^{50} -0.0641778 q^{52} -6.55438 q^{53} +2.34730 q^{55} +12.5175 q^{58} -0.204393 q^{59} +14.5817 q^{61} +4.06418 q^{62} +6.17024 q^{64} +0.433763 q^{65} +3.75877 q^{67} +1.36184 q^{68} +9.96585 q^{71} +0.120615 q^{73} -10.4534 q^{74} +0.268571 q^{76} +0.327696 q^{79} +10.7365 q^{80} -0.652704 q^{82} -3.35504 q^{83} -9.20439 q^{85} +1.81521 q^{86} +2.53209 q^{88} +4.65270 q^{89} -2.90167 q^{92} +11.7665 q^{94} -1.81521 q^{95} +13.1206 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{5} - 3 q^{8} + 3 q^{10} - 3 q^{11} + 3 q^{13} - 6 q^{16} + 3 q^{17} + 9 q^{19} - 6 q^{20} + 3 q^{25} - 9 q^{26} + 3 q^{29} + 9 q^{31} + 9 q^{32} - 6 q^{34} + 3 q^{40} - 9 q^{41} - 24 q^{46} + 3 q^{47}+ \cdots + 45 q^{97}+O(q^{100}) \) 3 * q - 6 * q^5 - 3 * q^8 + 3 * q^10 - 3 * q^11 + 3 * q^13 - 6 * q^16 + 3 * q^17 + 9 * q^19 - 6 * q^20 + 3 * q^25 - 9 * q^26 + 3 * q^29 + 9 * q^31 + 9 * q^32 - 6 * q^34 + 3 * q^40 - 9 * q^41 - 24 * q^46 + 3 * q^47 - 15 * q^50 + 9 * q^52 - 9 * q^53 + 6 * q^55 + 15 * q^58 + 12 * q^61 + 3 * q^62 - 3 * q^64 - 15 * q^65 + 21 * q^68 + 9 * q^71 + 6 * q^73 - 18 * q^74 - 9 * q^76 - 3 * q^79 + 27 * q^80 - 3 * q^82 + 15 * q^83 - 27 * q^85 + 9 * q^86 + 3 * q^88 + 15 * q^89 + 3 * q^92 - 9 * q^95 + 45 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more