LMFDB - Embedded newform 4600.2.a.bb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.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:	$36.7311849298$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.15529.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} - x + 2 $ x^4 - x^3 - 6*x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.73914$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.35292 q^{3} -1.14999 q^{7} +2.53622 q^{9} -2.32830 q^{11} -3.91376 q^{13} +3.47829 q^{17} -6.32830 q^{19} -2.70583 q^{21} -1.00000 q^{23} -1.09124 q^{27} +5.03913 q^{29} +1.11668 q^{31} -5.47829 q^{33} +3.00582 q^{37} -9.20874 q^{39} +1.50291 q^{41} -5.03413 q^{43} +2.76376 q^{47} -5.67752 q^{49} +8.18412 q^{51} -9.41167 q^{53} -14.8900 q^{57} -6.90507 q^{59} +3.95076 q^{61} -2.91663 q^{63} -0.705834 q^{67} -2.35292 q^{69} -7.42036 q^{71} -2.08255 q^{73} +2.67752 q^{77} -17.6907 q^{79} -10.1763 q^{81} -7.38335 q^{83} +11.8566 q^{87} -0.472471 q^{89} +4.50078 q^{91} +2.62746 q^{93} -8.82170 q^{97} -5.90507 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{7} + 6 q^{9} + q^{11} - 3 q^{13} - 2 q^{17} - 15 q^{19} + 8 q^{21} - 4 q^{23} - 27 q^{27} + q^{29} - 12 q^{31} - 6 q^{33} - 18 q^{37} - 3 q^{39} - 9 q^{41} + 9 q^{43} + 4 q^{47} - 3 q^{49} - 2 q^{51}+ \cdots - q^{99}+O(q^{100}) $ 4 * q + q^7 + 6 * q^9 + q^11 - 3 * q^13 - 2 * q^17 - 15 * q^19 + 8 * q^21 - 4 * q^23 - 27 * q^27 + q^29 - 12 * q^31 - 6 * q^33 - 18 * q^37 - 3 * q^39 - 9 * q^41 + 9 * q^43 + 4 * q^47 - 3 * q^49 - 2 * q^51 - 6 * q^57 - 5 * q^59 + 14 * q^61 - 39 * q^63 + 16 * q^67 - 2 * q^71 - 21 * q^73 - 9 * q^77 - 21 * q^79 + 44 * q^81 + 9 * q^83 - 17 * q^87 - 16 * q^89 + 33 * q^91 + 29 * q^93 - 40 * q^97 - q^99

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 $

$2$	0	0
$2$	0	0
$3$	2.35292	1.35846	0.679229	−	0.733927i	$-0.262314\pi$
$3$	2.35292	1.35846	0.679229	+	0.733927i	$0.262314\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.14999	−0.434656	−0.217328	−	0.976099i	$-0.569734\pi$
$7$	−1.14999	−0.434656	−0.217328	+	0.976099i	$0.569734\pi$
$8$	0	0
$9$	2.53622	0.845406
$10$	0	0
$11$	−2.32830	−0.702008	−0.351004	−	0.936374i	$-0.614160\pi$
$11$	−2.32830	−0.702008	−0.351004	+	0.936374i	$0.614160\pi$
$12$	0	0
$13$	−3.91376	−1.08548	−0.542740	−	0.839901i	$-0.682613\pi$
$13$	−3.91376	−1.08548	−0.542740	+	0.839901i	$0.682613\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.47829	0.843609	0.421804	−	0.906687i	$-0.361397\pi$
$17$	3.47829	0.843609	0.421804	+	0.906687i	$0.361397\pi$
$18$	0	0
$19$	−6.32830	−1.45181	−0.725905	−	0.687795i	$-0.758579\pi$
$19$	−6.32830	−1.45181	−0.725905	+	0.687795i	$0.758579\pi$
$20$	0	0
$21$	−2.70583	−0.590461
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.09124	−0.210009
$28$	0	0
$29$	5.03913	0.935742	0.467871	−	0.883797i	$-0.345021\pi$
$29$	5.03913	0.935742	0.467871	+	0.883797i	$0.345021\pi$
$30$	0	0
$31$	1.11668	0.200562	0.100281	−	0.994959i	$-0.468026\pi$
$31$	1.11668	0.200562	0.100281	+	0.994959i	$0.468026\pi$
$32$	0	0
$33$	−5.47829	−0.953647
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.00582	0.494153	0.247077	−	0.968996i	$-0.420530\pi$
$37$	3.00582	0.494153	0.247077	+	0.968996i	$0.420530\pi$
$38$	0	0
$39$	−9.20874	−1.47458
$40$	0	0
$41$	1.50291	0.234715	0.117357	−	0.993090i	$-0.462558\pi$
$41$	1.50291	0.234715	0.117357	+	0.993090i	$0.462558\pi$
$42$	0	0
$43$	−5.03413	−0.767698	−0.383849	−	0.923396i	$-0.625402\pi$
$43$	−5.03413	−0.767698	−0.383849	+	0.923396i	$0.625402\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.76376	0.403136	0.201568	−	0.979474i	$-0.435396\pi$
$47$	2.76376	0.403136	0.201568	+	0.979474i	$0.435396\pi$
$48$	0	0
$49$	−5.67752	−0.811074
$50$	0	0
$51$	8.18412	1.14601
$52$	0	0
$53$	−9.41167	−1.29279	−0.646396	−	0.763002i	$-0.723725\pi$
$53$	−9.41167	−1.29279	−0.646396	+	0.763002i	$0.723725\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−14.8900	−1.97222
$58$	0	0
$59$	−6.90507	−0.898963	−0.449482	−	0.893290i	$-0.648391\pi$
$59$	−6.90507	−0.898963	−0.449482	+	0.893290i	$0.648391\pi$
$60$	0	0
$61$	3.95076	0.505843	0.252921	−	0.967487i	$-0.418609\pi$
$61$	3.95076	0.505843	0.252921	+	0.967487i	$0.418609\pi$
$62$	0	0
$63$	−2.91663	−0.367461
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.705834	−0.0862313	−0.0431157	−	0.999070i	$-0.513728\pi$
$67$	−0.705834	−0.0862313	−0.0431157	+	0.999070i	$0.513728\pi$
$68$	0	0
$69$	−2.35292	−0.283258
$70$	0	0
$71$	−7.42036	−0.880634	−0.440317	−	0.897842i	$-0.645134\pi$
$71$	−7.42036	−0.880634	−0.440317	+	0.897842i	$0.645134\pi$
$72$	0	0
$73$	−2.08255	−0.243744	−0.121872	−	0.992546i	$-0.538890\pi$
$73$	−2.08255	−0.243744	−0.121872	+	0.992546i	$0.538890\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.67752	0.305132
$78$	0	0
$79$	−17.6907	−1.99036	−0.995181	−	0.0980562i	$-0.968738\pi$
$79$	−17.6907	−1.99036	−0.995181	+	0.0980562i	$0.968738\pi$
$80$	0	0
$81$	−10.1763	−1.13069
$82$	0	0
$83$	−7.38335	−0.810428	−0.405214	−	0.914222i	$-0.632803\pi$
$83$	−7.38335	−0.810428	−0.405214	+	0.914222i	$0.632803\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	11.8566	1.27117
$88$	0	0
$89$	−0.472471	−0.0500818	−0.0250409	−	0.999686i	$-0.507972\pi$
$89$	−0.472471	−0.0500818	−0.0250409	+	0.999686i	$0.507972\pi$
$90$	0	0
$91$	4.50078	0.471810
$92$	0	0
$93$	2.62746	0.272455
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.82170	−0.895707	−0.447854	−	0.894107i	$-0.647811\pi$
$97$	−8.82170	−0.895707	−0.447854	+	0.894107i	$0.647811\pi$
$98$	0	0
$99$	−5.90507	−0.593482
$100$	0	0
$101$	8.03413	0.799426	0.399713	−	0.916640i	$-0.369110\pi$
$101$	8.03413	0.799426	0.399713	+	0.916640i	$0.369110\pi$
$102$	0	0
$103$	4.62828	0.456038	0.228019	−	0.973657i	$-0.426775\pi$
$103$	4.62828	0.456038	0.228019	+	0.973657i	$0.426775\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−9.25656	−0.886617	−0.443309	−	0.896369i	$-0.646195\pi$
$109$	−9.25656	−0.886617	−0.443309	+	0.896369i	$0.646195\pi$
$110$	0	0
$111$	7.07244	0.671286
$112$	0	0
$113$	−12.8900	−1.21258	−0.606292	−	0.795242i	$-0.707344\pi$
$113$	−12.8900	−1.21258	−0.606292	+	0.795242i	$0.707344\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−9.92614	−0.917672
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−5.57904	−0.507185
$122$	0	0
$123$	3.53622	0.318850
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.11668	0.0990894	0.0495447	−	0.998772i	$-0.484223\pi$
$127$	1.11668	0.0990894	0.0495447	+	0.998772i	$0.484223\pi$
$128$	0	0
$129$	−11.8449	−1.04288
$130$	0	0
$131$	22.0153	1.92349	0.961744	−	0.273950i	$-0.0883303\pi$
$131$	22.0153	1.92349	0.961744	+	0.273950i	$0.0883303\pi$
$132$	0	0
$133$	7.27749	0.631038
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.70002	−0.145242	−0.0726211	−	0.997360i	$-0.523136\pi$
$137$	−1.70002	−0.145242	−0.0726211	+	0.997360i	$0.523136\pi$
$138$	0	0
$139$	−9.23624	−0.783407	−0.391704	−	0.920091i	$-0.628114\pi$
$139$	−9.23624	−0.783407	−0.391704	+	0.920091i	$0.628114\pi$
$140$	0	0
$141$	6.50291	0.547644
$142$	0	0
$143$	9.11238	0.762016
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−13.3587	−1.10181
$148$	0	0
$149$	12.1275	0.993523	0.496762	−	0.867887i	$-0.334522\pi$
$149$	12.1275	0.993523	0.496762	+	0.867887i	$0.334522\pi$
$150$	0	0
$151$	−12.0203	−0.978200	−0.489100	−	0.872228i	$-0.662675\pi$
$151$	−12.0203	−0.978200	−0.489100	+	0.872228i	$0.662675\pi$
$152$	0	0
$153$	8.82170	0.713192
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−7.05506	−0.563055	−0.281527	−	0.959553i	$-0.590841\pi$
$157$	−7.05506	−0.563055	−0.281527	+	0.959553i	$0.590841\pi$
$158$	0	0
$159$	−22.1449	−1.75620
$160$	0	0
$161$	1.14999	0.0906320
$162$	0	0
$163$	−2.82669	−0.221404	−0.110702	−	0.993854i	$-0.535310\pi$
$163$	−2.82669	−0.221404	−0.110702	+	0.993854i	$0.535310\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.44997	−0.344349	−0.172175	−	0.985066i	$-0.555079\pi$
$167$	−4.44997	−0.344349	−0.172175	+	0.985066i	$0.555079\pi$
$168$	0	0
$169$	2.31748	0.178268
$170$	0	0
$171$	−16.0499	−1.22737
$172$	0	0
$173$	21.5131	1.63561	0.817806	−	0.575494i	$-0.195190\pi$
$173$	21.5131	1.63561	0.817806	+	0.575494i	$0.195190\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−16.2470	−1.22120
$178$	0	0
$179$	7.19493	0.537775	0.268887	−	0.963172i	$-0.413344\pi$
$179$	7.19493	0.537775	0.268887	+	0.963172i	$0.413344\pi$
$180$	0	0
$181$	11.2942	0.839489	0.419744	−	0.907642i	$-0.362120\pi$
$181$	11.2942	0.839489	0.419744	+	0.907642i	$0.362120\pi$
$182$	0	0
$183$	9.29581	0.687166
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−8.09848	−0.592220
$188$	0	0
$189$	1.25492	0.0912818
$190$	0	0
$191$	−2.56904	−0.185890	−0.0929448	−	0.995671i	$-0.529628\pi$
$191$	−2.56904	−0.185890	−0.0929448	+	0.995671i	$0.529628\pi$
$192$	0	0
$193$	−14.1994	−1.02210	−0.511049	−	0.859551i	$-0.670743\pi$
$193$	−14.1994	−1.02210	−0.511049	+	0.859551i	$0.670743\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.21613	0.442881	0.221440	−	0.975174i	$-0.428924\pi$
$197$	6.21613	0.442881	0.221440	+	0.975174i	$0.428924\pi$
$198$	0	0
$199$	16.2907	1.15482	0.577408	−	0.816456i	$-0.304064\pi$
$199$	16.2907	1.15482	0.577408	+	0.816456i	$0.304064\pi$
$200$	0	0
$201$	−1.66077	−0.117142
$202$	0	0
$203$	−5.79495	−0.406726
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.53622	−0.176279
$208$	0	0
$209$	14.7341	1.01918
$210$	0	0
$211$	28.8865	1.98863	0.994314	−	0.106492i	$-0.0339620\pi$
$211$	28.8865	1.98863	0.994314	+	0.106492i	$0.0339620\pi$
$212$	0	0
$213$	−17.4595	−1.19630
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.28417	−0.0871754
$218$	0	0
$219$	−4.90007	−0.331116
$220$	0	0
$221$	−13.6132	−0.915721
$222$	0	0
$223$	−7.56166	−0.506366	−0.253183	−	0.967418i	$-0.581477\pi$
$223$	−7.56166	−0.506366	−0.253183	+	0.967418i	$0.581477\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.0683	−1.06649	−0.533244	−	0.845962i	$-0.679027\pi$
$227$	−16.0683	−1.06649	−0.533244	+	0.845962i	$0.679027\pi$
$228$	0	0
$229$	24.4175	1.61355	0.806776	−	0.590857i	$-0.201210\pi$
$229$	24.4175	1.61355	0.806776	+	0.590857i	$0.201210\pi$
$230$	0	0
$231$	6.29998	0.414508
$232$	0	0
$233$	−19.4703	−1.27554	−0.637771	−	0.770226i	$-0.720143\pi$
$233$	−19.4703	−1.27554	−0.637771	+	0.770226i	$0.720143\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−41.6248	−2.70382
$238$	0	0
$239$	13.8602	0.896543	0.448271	−	0.893898i	$-0.352040\pi$
$239$	13.8602	0.896543	0.448271	+	0.893898i	$0.352040\pi$
$240$	0	0
$241$	4.92756	0.317412	0.158706	−	0.987326i	$-0.449268\pi$
$241$	4.92756	0.317412	0.158706	+	0.987326i	$0.449268\pi$
$242$	0	0
$243$	−20.6702	−1.32599
$244$	0	0
$245$	0	0
$246$	0	0
$247$	24.7674	1.57591
$248$	0	0
$249$	−17.3724	−1.10093
$250$	0	0
$251$	9.51015	0.600275	0.300138	−	0.953896i	$-0.402967\pi$
$251$	9.51015	0.600275	0.300138	+	0.953896i	$0.402967\pi$
$252$	0	0
$253$	2.32830	0.146379
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−23.2355	−1.44939	−0.724695	−	0.689070i	$-0.758019\pi$
$257$	−23.2355	−1.44939	−0.724695	+	0.689070i	$0.758019\pi$
$258$	0	0
$259$	−3.45666	−0.214787
$260$	0	0
$261$	12.7803	0.791082
$262$	0	0
$263$	−18.0290	−1.11172	−0.555858	−	0.831277i	$-0.687610\pi$
$263$	−18.0290	−1.11172	−0.555858	+	0.831277i	$0.687610\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.11169	−0.0680340
$268$	0	0
$269$	27.4111	1.67128	0.835641	−	0.549276i	$-0.185096\pi$
$269$	27.4111	1.67128	0.835641	+	0.549276i	$0.185096\pi$
$270$	0	0
$271$	−6.24847	−0.379568	−0.189784	−	0.981826i	$-0.560779\pi$
$271$	−6.24847	−0.379568	−0.189784	+	0.981826i	$0.560779\pi$
$272$	0	0
$273$	10.5900	0.640934
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.40443	0.0843837	0.0421919	−	0.999110i	$-0.486566\pi$
$277$	1.40443	0.0843837	0.0421919	+	0.999110i	$0.486566\pi$
$278$	0	0
$279$	2.83215	0.169556
$280$	0	0
$281$	9.18994	0.548226	0.274113	−	0.961698i	$-0.411616\pi$
$281$	9.18994	0.548226	0.274113	+	0.961698i	$0.411616\pi$
$282$	0	0
$283$	18.0016	1.07009	0.535043	−	0.844825i	$-0.320295\pi$
$283$	18.0016	1.07009	0.535043	+	0.844825i	$0.320295\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.72833	−0.102020
$288$	0	0
$289$	−4.90152	−0.288325
$290$	0	0
$291$	−20.7567	−1.21678
$292$	0	0
$293$	−8.42323	−0.492090	−0.246045	−	0.969258i	$-0.579131\pi$
$293$	−8.42323	−0.492090	−0.246045	+	0.969258i	$0.579131\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.54073	0.147428
$298$	0	0
$299$	3.91376	0.226338
$300$	0	0
$301$	5.78921	0.333684
$302$	0	0
$303$	18.9036	1.08599
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	10.8900	0.619508
$310$	0	0
$311$	18.0849	1.02550	0.512750	−	0.858538i	$-0.328627\pi$
$311$	18.0849	1.02550	0.512750	+	0.858538i	$0.328627\pi$
$312$	0	0
$313$	22.4915	1.27129	0.635647	−	0.771980i	$-0.280733\pi$
$313$	22.4915	1.27129	0.635647	+	0.771980i	$0.280733\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.8443	0.609075	0.304537	−	0.952500i	$-0.401498\pi$
$317$	10.8443	0.609075	0.304537	+	0.952500i	$0.401498\pi$
$318$	0	0
$319$	−11.7326	−0.656898
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−22.0116	−1.22476
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−21.7799	−1.20443
$328$	0	0
$329$	−3.17830	−0.175226
$330$	0	0
$331$	4.20804	0.231295	0.115648	−	0.993290i	$-0.463106\pi$
$331$	4.20804	0.231295	0.115648	+	0.993290i	$0.463106\pi$
$332$	0	0
$333$	7.62341	0.417760
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.41748	0.131689	0.0658444	−	0.997830i	$-0.479026\pi$
$337$	2.41748	0.131689	0.0658444	+	0.997830i	$0.479026\pi$
$338$	0	0
$339$	−30.3290	−1.64724
$340$	0	0
$341$	−2.59996	−0.140796
$342$	0	0
$343$	14.5790	0.787194
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.45154	−0.292654	−0.146327	−	0.989236i	$-0.546745\pi$
$347$	−5.45154	−0.292654	−0.146327	+	0.989236i	$0.546745\pi$
$348$	0	0
$349$	5.03119	0.269313	0.134657	−	0.990892i	$-0.457007\pi$
$349$	5.03119	0.269313	0.134657	+	0.990892i	$0.457007\pi$
$350$	0	0
$351$	4.27085	0.227961
$352$	0	0
$353$	−3.44203	−0.183201	−0.0916005	−	0.995796i	$-0.529198\pi$
$353$	−3.44203	−0.183201	−0.0916005	+	0.995796i	$0.529198\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−9.41167	−0.498118
$358$	0	0
$359$	−1.62985	−0.0860201	−0.0430100	−	0.999075i	$-0.513695\pi$
$359$	−1.62985	−0.0860201	−0.0430100	+	0.999075i	$0.513695\pi$
$360$	0	0
$361$	21.0473	1.10775
$362$	0	0
$363$	−13.1270	−0.688989
$364$	0	0
$365$	0	0
$366$	0	0
$367$	33.1442	1.73012	0.865058	−	0.501672i	$-0.167282\pi$
$367$	33.1442	1.73012	0.865058	+	0.501672i	$0.167282\pi$
$368$	0	0
$369$	3.81170	0.198429
$370$	0	0
$371$	10.8233	0.561920
$372$	0	0
$373$	−0.988438	−0.0511794	−0.0255897	−	0.999673i	$-0.508146\pi$
$373$	−0.988438	−0.0511794	−0.0255897	+	0.999673i	$0.508146\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−19.7219	−1.01573
$378$	0	0
$379$	−30.0116	−1.54159	−0.770797	−	0.637081i	$-0.780142\pi$
$379$	−30.0116	−1.54159	−0.770797	+	0.637081i	$0.780142\pi$
$380$	0	0
$381$	2.62746	0.134609
$382$	0	0
$383$	5.79920	0.296325	0.148163	−	0.988963i	$-0.452664\pi$
$383$	5.79920	0.296325	0.148163	+	0.988963i	$0.452664\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−12.7677	−0.649016
$388$	0	0
$389$	−0.0173782	−0.000881108 0	−0.000440554	−	1.00000i	$-0.500140\pi$
$389$	−0.0173782	−0.000881108 0	−0.000440554		1.00000i	$0.500140\pi$
$390$	0	0
$391$	−3.47829	−0.175905
$392$	0	0
$393$	51.8002	2.61298
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−10.0414	−0.503963	−0.251982	−	0.967732i	$-0.581082\pi$
$397$	−10.0414	−0.503963	−0.251982	+	0.967732i	$0.581082\pi$
$398$	0	0
$399$	17.1233	0.857238
$400$	0	0
$401$	13.5639	0.677350	0.338675	−	0.940903i	$-0.390021\pi$
$401$	13.5639	0.677350	0.338675	+	0.940903i	$0.390021\pi$
$402$	0	0
$403$	−4.37042	−0.217706
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.99843	−0.346899
$408$	0	0
$409$	35.9320	1.77672	0.888362	−	0.459143i	$-0.151844\pi$
$409$	35.9320	1.77672	0.888362	+	0.459143i	$0.151844\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	0	0
$413$	7.94077	0.390740
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−21.7321	−1.06423
$418$	0	0
$419$	−23.4516	−1.14569	−0.572843	−	0.819665i	$-0.694160\pi$
$419$	−23.4516	−1.14569	−0.572843	+	0.819665i	$0.694160\pi$
$420$	0	0
$421$	19.4043	0.945707	0.472853	−	0.881141i	$-0.343224\pi$
$421$	19.4043	0.945707	0.472853	+	0.881141i	$0.343224\pi$
$422$	0	0
$423$	7.00951	0.340814
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.54334	−0.219868
$428$	0	0
$429$	21.4407	1.03517
$430$	0	0
$431$	1.95076	0.0939647	0.0469824	−	0.998896i	$-0.485040\pi$
$431$	1.95076	0.0939647	0.0469824	+	0.998896i	$0.485040\pi$
$432$	0	0
$433$	−4.24336	−0.203923	−0.101961	−	0.994788i	$-0.532512\pi$
$433$	−4.24336	−0.203923	−0.101961	+	0.994788i	$0.532512\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.32830	0.302723
$438$	0	0
$439$	3.10799	0.148336	0.0741682	−	0.997246i	$-0.476370\pi$
$439$	3.10799	0.148336	0.0741682	+	0.997246i	$0.476370\pi$
$440$	0	0
$441$	−14.3994	−0.685687
$442$	0	0
$443$	10.1652	0.482963	0.241481	−	0.970405i	$-0.422367\pi$
$443$	10.1652	0.482963	0.241481	+	0.970405i	$0.422367\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	28.5350	1.34966
$448$	0	0
$449$	12.2659	0.578861	0.289431	−	0.957199i	$-0.406534\pi$
$449$	12.2659	0.578861	0.289431	+	0.957199i	$0.406534\pi$
$450$	0	0
$451$	−3.49922	−0.164772
$452$	0	0
$453$	−28.2828	−1.32884
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.0016	−1.02919	−0.514597	−	0.857432i	$-0.672058\pi$
$457$	−22.0016	−1.02919	−0.514597	+	0.857432i	$0.672058\pi$
$458$	0	0
$459$	−3.79565	−0.177166
$460$	0	0
$461$	31.4111	1.46296	0.731479	−	0.681863i	$-0.238830\pi$
$461$	31.4111	1.46296	0.731479	+	0.681863i	$0.238830\pi$
$462$	0	0
$463$	−24.4764	−1.13751	−0.568757	−	0.822506i	$-0.692575\pi$
$463$	−24.4764	−1.13751	−0.568757	+	0.822506i	$0.692575\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14.5050	−0.671213	−0.335606	−	0.942002i	$-0.608941\pi$
$467$	−14.5050	−0.671213	−0.335606	+	0.942002i	$0.608941\pi$
$468$	0	0
$469$	0.811703	0.0374809
$470$	0	0
$471$	−16.6000	−0.764886
$472$	0	0
$473$	11.7209	0.538930
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−23.8700	−1.09293
$478$	0	0
$479$	−27.8472	−1.27237	−0.636186	−	0.771536i	$-0.719489\pi$
$479$	−27.8472	−1.27237	−0.636186	+	0.771536i	$0.719489\pi$
$480$	0	0
$481$	−11.7640	−0.536394
$482$	0	0
$483$	2.70583	0.123120
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7.57547	−0.343277	−0.171639	−	0.985160i	$-0.554906\pi$
$487$	−7.57547	−0.343277	−0.171639	+	0.985160i	$0.554906\pi$
$488$	0	0
$489$	−6.65097	−0.300767
$490$	0	0
$491$	−7.43773	−0.335660	−0.167830	−	0.985816i	$-0.553676\pi$
$491$	−7.43773	−0.335660	−0.167830	+	0.985816i	$0.553676\pi$
$492$	0	0
$493$	17.5275	0.789400
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.53335	0.382773
$498$	0	0
$499$	26.4602	1.18452	0.592260	−	0.805747i	$-0.298236\pi$
$499$	26.4602	1.18452	0.592260	+	0.805747i	$0.298236\pi$
$500$	0	0
$501$	−10.4704	−0.467784
$502$	0	0
$503$	−22.9385	−1.02278	−0.511389	−	0.859350i	$-0.670869\pi$
$503$	−22.9385	−1.02278	−0.511389	+	0.859350i	$0.670869\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5.45285	0.242169
$508$	0	0
$509$	−34.5603	−1.53186	−0.765929	−	0.642925i	$-0.777721\pi$
$509$	−34.5603	−1.53186	−0.765929	+	0.642925i	$0.777721\pi$
$510$	0	0
$511$	2.39492	0.105945
$512$	0	0
$513$	6.90569	0.304894
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−6.43486	−0.283005
$518$	0	0
$519$	50.6186	2.22191
$520$	0	0
$521$	−15.2202	−0.666807	−0.333404	−	0.942784i	$-0.608197\pi$
$521$	−15.2202	−0.666807	−0.333404	+	0.942784i	$0.608197\pi$
$522$	0	0
$523$	−18.0892	−0.790985	−0.395492	−	0.918469i	$-0.629426\pi$
$523$	−18.0892	−0.790985	−0.395492	+	0.918469i	$0.629426\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.88414	0.169196
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−17.5128	−0.759989
$532$	0	0
$533$	−5.88202	−0.254778
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.9291	0.730544
$538$	0	0
$539$	13.2189	0.569380
$540$	0	0
$541$	43.1203	1.85389	0.926944	−	0.375200i	$-0.122426\pi$
$541$	43.1203	1.85389	0.926944	+	0.375200i	$0.122426\pi$
$542$	0	0
$543$	26.5742	1.14041
$544$	0	0
$545$	0	0
$546$	0	0
$547$	37.6045	1.60785	0.803926	−	0.594730i	$-0.202741\pi$
$547$	37.6045	1.60785	0.803926	+	0.594730i	$0.202741\pi$
$548$	0	0
$549$	10.0200	0.427643
$550$	0	0
$551$	−31.8891	−1.35852
$552$	0	0
$553$	20.3442	0.865122
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.1085	−0.597795	−0.298898	−	0.954285i	$-0.596619\pi$
$557$	−14.1085	−0.597795	−0.298898	+	0.954285i	$0.596619\pi$
$558$	0	0
$559$	19.7024	0.833321
$560$	0	0
$561$	−19.0551	−0.804505
$562$	0	0
$563$	−38.5675	−1.62543	−0.812714	−	0.582663i	$-0.802011\pi$
$563$	−38.5675	−1.62543	−0.812714	+	0.582663i	$0.802011\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	11.7026	0.491463
$568$	0	0
$569$	−1.62316	−0.0680464	−0.0340232	−	0.999421i	$-0.510832\pi$
$569$	−1.62316	−0.0680464	−0.0340232	+	0.999421i	$0.510832\pi$
$570$	0	0
$571$	10.2942	0.430800	0.215400	−	0.976526i	$-0.430894\pi$
$571$	10.2942	0.430800	0.215400	+	0.976526i	$0.430894\pi$
$572$	0	0
$573$	−6.04475	−0.252523
$574$	0	0
$575$	0	0
$576$	0	0
$577$	36.2278	1.50818	0.754091	−	0.656770i	$-0.228078\pi$
$577$	36.2278	1.50818	0.754091	+	0.656770i	$0.228078\pi$
$578$	0	0
$579$	−33.4101	−1.38848
$580$	0	0
$581$	8.49079	0.352257
$582$	0	0
$583$	21.9131	0.907550
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−34.1626	−1.41004	−0.705020	−	0.709187i	$-0.749062\pi$
$587$	−34.1626	−1.41004	−0.705020	+	0.709187i	$0.749062\pi$
$588$	0	0
$589$	−7.06669	−0.291178
$590$	0	0
$591$	14.6260	0.601635
$592$	0	0
$593$	19.7326	0.810320	0.405160	−	0.914246i	$-0.367216\pi$
$593$	19.7326	0.810320	0.405160	+	0.914246i	$0.367216\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	38.3306	1.56877
$598$	0	0
$599$	15.5500	0.635357	0.317678	−	0.948199i	$-0.397097\pi$
$599$	15.5500	0.635357	0.317678	+	0.948199i	$0.397097\pi$
$600$	0	0
$601$	16.7182	0.681950	0.340975	−	0.940072i	$-0.389243\pi$
$601$	16.7182	0.681950	0.340975	+	0.940072i	$0.389243\pi$
$602$	0	0
$603$	−1.79015	−0.0729005
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−26.4784	−1.07472	−0.537362	−	0.843352i	$-0.680579\pi$
$607$	−26.4784	−1.07472	−0.537362	+	0.843352i	$0.680579\pi$
$608$	0	0
$609$	−13.6350	−0.552520
$610$	0	0
$611$	−10.8167	−0.437597
$612$	0	0
$613$	18.1667	0.733748	0.366874	−	0.930271i	$-0.380428\pi$
$613$	18.1667	0.733748	0.366874	+	0.930271i	$0.380428\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	43.4540	1.74939	0.874695	−	0.484674i	$-0.161062\pi$
$617$	43.4540	1.74939	0.874695	+	0.484674i	$0.161062\pi$
$618$	0	0
$619$	−25.3841	−1.02027	−0.510136	−	0.860094i	$-0.670405\pi$
$619$	−25.3841	−1.02027	−0.510136	+	0.860094i	$0.670405\pi$
$620$	0	0
$621$	1.09124	0.0437900
$622$	0	0
$623$	0.543338	0.0217684
$624$	0	0
$625$	0	0
$626$	0	0
$627$	34.6682	1.38452
$628$	0	0
$629$	10.4551	0.416872
$630$	0	0
$631$	−43.2646	−1.72234	−0.861169	−	0.508319i	$-0.830267\pi$
$631$	−43.2646	−1.72234	−0.861169	+	0.508319i	$0.830267\pi$
$632$	0	0
$633$	67.9675	2.70146
$634$	0	0
$635$	0	0
$636$	0	0
$637$	22.2204	0.880405
$638$	0	0
$639$	−18.8196	−0.744494
$640$	0	0
$641$	20.6508	0.815659	0.407830	−	0.913058i	$-0.366286\pi$
$641$	20.6508	0.815659	0.407830	+	0.913058i	$0.366286\pi$
$642$	0	0
$643$	−40.2357	−1.58674	−0.793371	−	0.608739i	$-0.791676\pi$
$643$	−40.2357	−1.58674	−0.793371	+	0.608739i	$0.791676\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.11511	−0.161782	−0.0808909	−	0.996723i	$-0.525777\pi$
$647$	−4.11511	−0.161782	−0.0808909	+	0.996723i	$0.525777\pi$
$648$	0	0
$649$	16.0770	0.631079
$650$	0	0
$651$	−3.02155	−0.118424
$652$	0	0
$653$	−6.74914	−0.264114	−0.132057	−	0.991242i	$-0.542158\pi$
$653$	−6.74914	−0.264114	−0.132057	+	0.991242i	$0.542158\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5.28180	−0.206063
$658$	0	0
$659$	15.7500	0.613531	0.306766	−	0.951785i	$-0.400753\pi$
$659$	15.7500	0.613531	0.306766	+	0.951785i	$0.400753\pi$
$660$	0	0
$661$	7.51590	0.292334	0.146167	−	0.989260i	$-0.453306\pi$
$661$	7.51590	0.292334	0.146167	+	0.989260i	$0.453306\pi$
$662$	0	0
$663$	−32.0307	−1.24397
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.03913	−0.195116
$668$	0	0
$669$	−17.7920	−0.687877
$670$	0	0
$671$	−9.19854	−0.355106
$672$	0	0
$673$	−7.14630	−0.275470	−0.137735	−	0.990469i	$-0.543982\pi$
$673$	−7.14630	−0.275470	−0.137735	+	0.990469i	$0.543982\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0.568102	0.0218339	0.0109170	−	0.999940i	$-0.496525\pi$
$677$	0.568102	0.0218339	0.0109170	+	0.999940i	$0.496525\pi$
$678$	0	0
$679$	10.1449	0.389324
$680$	0	0
$681$	−37.8073	−1.44878
$682$	0	0
$683$	−4.52671	−0.173210	−0.0866048	−	0.996243i	$-0.527602\pi$
$683$	−4.52671	−0.173210	−0.0866048	+	0.996243i	$0.527602\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	57.4523	2.19194
$688$	0	0
$689$	36.8350	1.40330
$690$	0	0
$691$	−15.0499	−0.572527	−0.286263	−	0.958151i	$-0.592413\pi$
$691$	−15.0499	−0.572527	−0.286263	+	0.958151i	$0.592413\pi$
$692$	0	0
$693$	6.79077	0.257960
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.22755	0.198007
$698$	0	0
$699$	−45.8120	−1.73277
$700$	0	0
$701$	−34.3696	−1.29812	−0.649062	−	0.760736i	$-0.724838\pi$
$701$	−34.3696	−1.29812	−0.649062	+	0.760736i	$0.724838\pi$
$702$	0	0
$703$	−19.0217	−0.717417
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.23918	−0.347475
$708$	0	0
$709$	−30.1391	−1.13190	−0.565949	−	0.824440i	$-0.691490\pi$
$709$	−30.1391	−1.13190	−0.565949	+	0.824440i	$0.691490\pi$
$710$	0	0
$711$	−44.8675	−1.68266
$712$	0	0
$713$	−1.11668	−0.0418200
$714$	0	0
$715$	0	0
$716$	0	0
$717$	32.6119	1.21792
$718$	0	0
$719$	13.6015	0.507252	0.253626	−	0.967302i	$-0.418377\pi$
$719$	13.6015	0.507252	0.253626	+	0.967302i	$0.418377\pi$
$720$	0	0
$721$	−5.32248	−0.198220
$722$	0	0
$723$	11.5941	0.431191
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−31.3146	−1.16139	−0.580697	−	0.814120i	$-0.697220\pi$
$727$	−31.3146	−1.16139	−0.580697	+	0.814120i	$0.697220\pi$
$728$	0	0
$729$	−18.1064	−0.670607
$730$	0	0
$731$	−17.5102	−0.647636
$732$	0	0
$733$	−4.76985	−0.176178	−0.0880892	−	0.996113i	$-0.528076\pi$
$733$	−4.76985	−0.176178	−0.0880892	+	0.996113i	$0.528076\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.64339	0.0605351
$738$	0	0
$739$	2.65160	0.0975405	0.0487703	−	0.998810i	$-0.484470\pi$
$739$	2.65160	0.0975405	0.0487703	+	0.998810i	$0.484470\pi$
$740$	0	0
$741$	58.2756	2.14081
$742$	0	0
$743$	−35.8225	−1.31420	−0.657099	−	0.753804i	$-0.728217\pi$
$743$	−35.8225	−1.31420	−0.657099	+	0.753804i	$0.728217\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−18.7258	−0.685141
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−22.1834	−0.809485	−0.404742	−	0.914431i	$-0.632639\pi$
$751$	−22.1834	−0.809485	−0.404742	+	0.914431i	$0.632639\pi$
$752$	0	0
$753$	22.3766	0.815448
$754$	0	0
$755$	0	0
$756$	0	0
$757$	20.9318	0.760780	0.380390	−	0.924826i	$-0.375790\pi$
$757$	20.9318	0.760780	0.380390	+	0.924826i	$0.375790\pi$
$758$	0	0
$759$	5.47829	0.198849
$760$	0	0
$761$	−51.5641	−1.86920	−0.934599	−	0.355703i	$-0.884241\pi$
$761$	−51.5641	−1.86920	−0.934599	+	0.355703i	$0.884241\pi$
$762$	0	0
$763$	10.6450	0.385373
$764$	0	0
$765$	0	0
$766$	0	0
$767$	27.0247	0.975807
$768$	0	0
$769$	−0.913079	−0.0329265	−0.0164632	−	0.999864i	$-0.505241\pi$
$769$	−0.913079	−0.0329265	−0.0164632	+	0.999864i	$0.505241\pi$
$770$	0	0
$771$	−54.6712	−1.96893
$772$	0	0
$773$	−21.4709	−0.772255	−0.386127	−	0.922445i	$-0.626187\pi$
$773$	−21.4709	−0.772255	−0.386127	+	0.922445i	$0.626187\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−8.13324	−0.291778
$778$	0	0
$779$	−9.51085	−0.340761
$780$	0	0
$781$	17.2768	0.618212
$782$	0	0
$783$	−5.49890	−0.196515
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−51.0446	−1.81954	−0.909771	−	0.415111i	$-0.863743\pi$
$787$	−51.0446	−1.81954	−0.909771	+	0.415111i	$0.863743\pi$
$788$	0	0
$789$	−42.4208	−1.51022
$790$	0	0
$791$	14.8233	0.527057
$792$	0	0
$793$	−15.4623	−0.549082
$794$	0	0
$795$	0	0
$796$	0	0
$797$	50.2250	1.77906	0.889531	−	0.456876i	$-0.151032\pi$
$797$	50.2250	1.77906	0.889531	+	0.456876i	$0.151032\pi$
$798$	0	0
$799$	9.61317	0.340089
$800$	0	0
$801$	−1.19829	−0.0423395
$802$	0	0
$803$	4.84880	0.171110
$804$	0	0
$805$	0	0
$806$	0	0
$807$	64.4960	2.27036
$808$	0	0
$809$	−38.5218	−1.35435	−0.677177	−	0.735820i	$-0.736797\pi$
$809$	−38.5218	−1.35435	−0.677177	+	0.735820i	$0.736797\pi$
$810$	0	0
$811$	42.0912	1.47802	0.739011	−	0.673694i	$-0.235293\pi$
$811$	42.0912	1.47802	0.739011	+	0.673694i	$0.235293\pi$
$812$	0	0
$813$	−14.7021	−0.515627
$814$	0	0
$815$	0	0
$816$	0	0
$817$	31.8575	1.11455
$818$	0	0
$819$	11.4150	0.398871
$820$	0	0
$821$	−19.4146	−0.677575	−0.338788	−	0.940863i	$-0.610017\pi$
$821$	−19.4146	−0.677575	−0.338788	+	0.940863i	$0.610017\pi$
$822$	0	0
$823$	22.0646	0.769122	0.384561	−	0.923099i	$-0.374353\pi$
$823$	22.0646	0.769122	0.384561	+	0.923099i	$0.374353\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	40.3792	1.40412	0.702061	−	0.712117i	$-0.252263\pi$
$827$	40.3792	1.40412	0.702061	+	0.712117i	$0.252263\pi$
$828$	0	0
$829$	−20.5682	−0.714362	−0.357181	−	0.934035i	$-0.616262\pi$
$829$	−20.5682	−0.714362	−0.357181	+	0.934035i	$0.616262\pi$
$830$	0	0
$831$	3.30450	0.114632
$832$	0	0
$833$	−19.7480	−0.684229
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.21857	−0.0421199
$838$	0	0
$839$	−30.2414	−1.04405	−0.522025	−	0.852930i	$-0.674823\pi$
$839$	−30.2414	−1.04405	−0.522025	+	0.852930i	$0.674823\pi$
$840$	0	0
$841$	−3.60721	−0.124386
$842$	0	0
$843$	21.6232	0.744741
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.41584	0.220451
$848$	0	0
$849$	42.3564	1.45367
$850$	0	0
$851$	−3.00582	−0.103038
$852$	0	0
$853$	−32.0590	−1.09768	−0.548839	−	0.835928i	$-0.684930\pi$
$853$	−32.0590	−1.09768	−0.548839	+	0.835928i	$0.684930\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−53.6012	−1.83098	−0.915491	−	0.402338i	$-0.868198\pi$
$857$	−53.6012	−1.83098	−0.915491	+	0.402338i	$0.868198\pi$
$858$	0	0
$859$	3.33677	0.113849	0.0569246	−	0.998378i	$-0.481871\pi$
$859$	3.33677	0.113849	0.0569246	+	0.998378i	$0.481871\pi$
$860$	0	0
$861$	−4.06662	−0.138590
$862$	0	0
$863$	−53.5226	−1.82193	−0.910966	−	0.412482i	$-0.864662\pi$
$863$	−53.5226	−1.82193	−0.910966	+	0.412482i	$0.864662\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−11.5329	−0.391677
$868$	0	0
$869$	41.1892	1.39725
$870$	0	0
$871$	2.76246	0.0936024
$872$	0	0
$873$	−22.3737	−0.757236
$874$	0	0
$875$	0	0
$876$	0	0
$877$	51.2457	1.73044	0.865222	−	0.501389i	$-0.167177\pi$
$877$	51.2457	1.73044	0.865222	+	0.501389i	$0.167177\pi$
$878$	0	0
$879$	−19.8192	−0.668484
$880$	0	0
$881$	−40.3392	−1.35906	−0.679532	−	0.733646i	$-0.737817\pi$
$881$	−40.3392	−1.35906	−0.679532	+	0.733646i	$0.737817\pi$
$882$	0	0
$883$	−29.3763	−0.988592	−0.494296	−	0.869294i	$-0.664574\pi$
$883$	−29.3763	−0.988592	−0.494296	+	0.869294i	$0.664574\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	57.1776	1.91984	0.959919	−	0.280278i	$-0.0904268\pi$
$887$	57.1776	1.91984	0.959919	+	0.280278i	$0.0904268\pi$
$888$	0	0
$889$	−1.28417	−0.0430698
$890$	0	0
$891$	23.6933	0.793756
$892$	0	0
$893$	−17.4899	−0.585278
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9.20874	0.307471
$898$	0	0
$899$	5.62710	0.187674
$900$	0	0
$901$	−32.7365	−1.09061
$902$	0	0
$903$	13.6215	0.453296
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−46.8357	−1.55515	−0.777576	−	0.628789i	$-0.783551\pi$
$907$	−46.8357	−1.55515	−0.777576	+	0.628789i	$0.783551\pi$
$908$	0	0
$909$	20.3763	0.675839
$910$	0	0
$911$	−16.7225	−0.554042	−0.277021	−	0.960864i	$-0.589347\pi$
$911$	−16.7225	−0.554042	−0.277021	+	0.960864i	$0.589347\pi$
$912$	0	0
$913$	17.1906	0.568927
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−25.3174	−0.836055
$918$	0	0
$919$	−28.7915	−0.949743	−0.474872	−	0.880055i	$-0.657505\pi$
$919$	−28.7915	−0.949743	−0.474872	+	0.880055i	$0.657505\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	29.0415	0.955911
$924$	0	0
$925$	0	0
$926$	0	0
$927$	11.7383	0.385537
$928$	0	0
$929$	−21.1756	−0.694750	−0.347375	−	0.937726i	$-0.612927\pi$
$929$	−21.1756	−0.694750	−0.347375	+	0.937726i	$0.612927\pi$
$930$	0	0
$931$	35.9290	1.17753
$932$	0	0
$933$	42.5522	1.39310
$934$	0	0
$935$	0	0
$936$	0	0
$937$	29.5408	0.965056	0.482528	−	0.875881i	$-0.339719\pi$
$937$	29.5408	0.965056	0.482528	+	0.875881i	$0.339719\pi$
$938$	0	0
$939$	52.9206	1.72700
$940$	0	0
$941$	2.20314	0.0718203	0.0359101	−	0.999355i	$-0.488567\pi$
$941$	2.20314	0.0718203	0.0359101	+	0.999355i	$0.488567\pi$
$942$	0	0
$943$	−1.50291	−0.0489414
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24.5548	−0.797922	−0.398961	−	0.916968i	$-0.630629\pi$
$947$	−24.5548	−0.797922	−0.398961	+	0.916968i	$0.630629\pi$
$948$	0	0
$949$	8.15060	0.264580
$950$	0	0
$951$	25.5157	0.827402
$952$	0	0
$953$	38.5697	1.24940	0.624698	−	0.780866i	$-0.285222\pi$
$953$	38.5697	1.24940	0.624698	+	0.780866i	$0.285222\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−27.6058	−0.892368
$958$	0	0
$959$	1.95501	0.0631304
$960$	0	0
$961$	−29.7530	−0.959775
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.85650	0.156175	0.0780873	−	0.996947i	$-0.475119\pi$
$967$	4.85650	0.156175	0.0780873	+	0.996947i	$0.475119\pi$
$968$	0	0
$969$	−51.7915	−1.66378
$970$	0	0
$971$	−10.4384	−0.334985	−0.167492	−	0.985873i	$-0.553567\pi$
$971$	−10.4384	−0.334985	−0.167492	+	0.985873i	$0.553567\pi$
$972$	0	0
$973$	10.6216	0.340513
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−39.9985	−1.27967	−0.639833	−	0.768514i	$-0.720996\pi$
$977$	−39.9985	−1.27967	−0.639833	+	0.768514i	$0.720996\pi$
$978$	0	0
$979$	1.10005	0.0351578
$980$	0	0
$981$	−23.4766	−0.749552
$982$	0	0
$983$	−9.68073	−0.308767	−0.154384	−	0.988011i	$-0.549339\pi$
$983$	−9.68073	−0.308767	−0.154384	+	0.988011i	$0.549339\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−7.47829	−0.238037
$988$	0	0
$989$	5.03413	0.160076
$990$	0	0
$991$	−36.6733	−1.16497	−0.582484	−	0.812842i	$-0.697919\pi$
$991$	−36.6733	−1.16497	−0.582484	+	0.812842i	$0.697919\pi$
$992$	0	0
$993$	9.90118	0.314204
$994$	0	0
$995$	0	0
$996$	0	0
$997$	46.2901	1.46602	0.733010	−	0.680217i	$-0.238115\pi$
$997$	46.2901	1.46602	0.733010	+	0.680217i	$0.238115\pi$
$998$	0	0
$999$	−3.28007	−0.103777

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.a.bb.1.4	yes	4
4.3	odd	2		9200.2.a.cn.1.1		4
5.2	odd	4		4600.2.e.t.4049.2		8
5.3	odd	4		4600.2.e.t.4049.7		8
5.4	even	2		4600.2.a.ba.1.1	✓	4
20.19	odd	2		9200.2.a.cp.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.ba.1.1	✓	4	5.4	even	2
4600.2.a.bb.1.4	yes	4	1.1	even	1	trivial
4600.2.e.t.4049.2		8	5.2	odd	4
4600.2.e.t.4049.7		8	5.3	odd	4
9200.2.a.cn.1.1		4	4.3	odd	2
9200.2.a.cp.1.4		4	20.19	odd	2

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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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)

\(f(q)\)	\(=\)	\(q+2.35292 q^{3} -1.14999 q^{7} +2.53622 q^{9} -2.32830 q^{11} -3.91376 q^{13} +3.47829 q^{17} -6.32830 q^{19} -2.70583 q^{21} -1.00000 q^{23} -1.09124 q^{27} +5.03913 q^{29} +1.11668 q^{31} -5.47829 q^{33} +3.00582 q^{37} -9.20874 q^{39} +1.50291 q^{41} -5.03413 q^{43} +2.76376 q^{47} -5.67752 q^{49} +8.18412 q^{51} -9.41167 q^{53} -14.8900 q^{57} -6.90507 q^{59} +3.95076 q^{61} -2.91663 q^{63} -0.705834 q^{67} -2.35292 q^{69} -7.42036 q^{71} -2.08255 q^{73} +2.67752 q^{77} -17.6907 q^{79} -10.1763 q^{81} -7.38335 q^{83} +11.8566 q^{87} -0.472471 q^{89} +4.50078 q^{91} +2.62746 q^{93} -8.82170 q^{97} -5.90507 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{7} + 6 q^{9} + q^{11} - 3 q^{13} - 2 q^{17} - 15 q^{19} + 8 q^{21} - 4 q^{23} - 27 q^{27} + q^{29} - 12 q^{31} - 6 q^{33} - 18 q^{37} - 3 q^{39} - 9 q^{41} + 9 q^{43} + 4 q^{47} - 3 q^{49} - 2 q^{51}+ \cdots - q^{99}+O(q^{100}) \) 4 * q + q^7 + 6 * q^9 + q^11 - 3 * q^13 - 2 * q^17 - 15 * q^19 + 8 * q^21 - 4 * q^23 - 27 * q^27 + q^29 - 12 * q^31 - 6 * q^33 - 18 * q^37 - 3 * q^39 - 9 * q^41 + 9 * q^43 + 4 * q^47 - 3 * q^49 - 2 * q^51 - 6 * q^57 - 5 * q^59 + 14 * q^61 - 39 * q^63 + 16 * q^67 - 2 * q^71 - 21 * q^73 - 9 * q^77 - 21 * q^79 + 44 * q^81 + 9 * q^83 - 17 * q^87 - 16 * q^89 + 33 * q^91 + 29 * q^93 - 40 * q^97 - q^99

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