LMFDB - Embedded newform 5800.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5800,2,Mod(1,5800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5800, 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("5800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5800 = 2^{3} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5800.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:	$46.3132331723$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1160)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	5800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.93543 q^{3} +1.25410 q^{7} +5.61676 q^{9} +2.50820 q^{11} +2.93543 q^{13} -7.12497 q^{17} +4.85446 q^{19} -3.68133 q^{21} -1.57277 q^{23} -7.68133 q^{27} -1.00000 q^{29} -4.61676 q^{31} -7.36266 q^{33} -9.87086 q^{37} -8.61676 q^{39} +0.508203 q^{41} -1.38324 q^{43} +1.36266 q^{47} -5.42723 q^{49} +20.9149 q^{51} -2.23769 q^{53} -14.2499 q^{57} -11.6608 q^{59} +3.41082 q^{61} +7.04399 q^{63} +6.72532 q^{67} +4.61676 q^{69} +14.7253 q^{71} +12.3585 q^{73} +3.14554 q^{77} +3.91903 q^{79} +5.69774 q^{81} +1.87086 q^{83} +2.93543 q^{87} -11.8709 q^{89} +3.68133 q^{91} +13.5522 q^{93} -5.31450 q^{97} +14.0880 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 3 q^{7} + 2 q^{9} + 6 q^{11} + q^{13} - 5 q^{17} + 2 q^{19} - 4 q^{21} - 11 q^{23} - 16 q^{27} - 3 q^{29} + q^{31} - 8 q^{33} - 14 q^{37} - 11 q^{39} - 19 q^{43} - 10 q^{47} - 10 q^{49}+ \cdots + 9 q^{97}+O(q^{100}) $ 3 * q - q^3 + 3 * q^7 + 2 * q^9 + 6 * q^11 + q^13 - 5 * q^17 + 2 * q^19 - 4 * q^21 - 11 * q^23 - 16 * q^27 - 3 * q^29 + q^31 - 8 * q^33 - 14 * q^37 - 11 * q^39 - 19 * q^43 - 10 * q^47 - 10 * q^49 + 26 * q^51 - 9 * q^53 - 10 * q^57 + q^59 + 7 * q^61 - 8 * q^67 - q^69 + 16 * q^71 - 9 * q^73 + 22 * q^77 + 7 * q^79 + 7 * q^81 - 10 * q^83 + q^87 - 20 * q^89 + 4 * q^91 + 18 * q^93 + 9 * 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 $

$2$	0	0
$2$	0	0
$3$	−2.93543	−1.69477	−0.847386	−	0.530977i	$-0.821825\pi$
$3$	−2.93543	−1.69477	−0.847386	+	0.530977i	$0.821825\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.25410	0.474006	0.237003	−	0.971509i	$-0.423835\pi$
$7$	1.25410	0.474006	0.237003	+	0.971509i	$0.423835\pi$
$8$	0	0
$9$	5.61676	1.87225
$10$	0	0
$11$	2.50820	0.756252	0.378126	−	0.925754i	$-0.376569\pi$
$11$	2.50820	0.756252	0.378126	+	0.925754i	$0.376569\pi$
$12$	0	0
$13$	2.93543	0.814142	0.407071	−	0.913396i	$-0.366550\pi$
$13$	2.93543	0.814142	0.407071	+	0.913396i	$0.366550\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.12497	−1.72806	−0.864029	−	0.503442i	$-0.832067\pi$
$17$	−7.12497	−1.72806	−0.864029	+	0.503442i	$0.832067\pi$
$18$	0	0
$19$	4.85446	1.11369	0.556845	−	0.830617i	$-0.312012\pi$
$19$	4.85446	1.11369	0.556845	+	0.830617i	$0.312012\pi$
$20$	0	0
$21$	−3.68133	−0.803332
$22$	0	0
$23$	−1.57277	−0.327945	−0.163973	−	0.986465i	$-0.552431\pi$
$23$	−1.57277	−0.327945	−0.163973	+	0.986465i	$0.552431\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−7.68133	−1.47827
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−4.61676	−0.829195	−0.414598	−	0.910005i	$-0.636078\pi$
$31$	−4.61676	−0.829195	−0.414598	+	0.910005i	$0.636078\pi$
$32$	0	0
$33$	−7.36266	−1.28167
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.87086	−1.62276	−0.811380	−	0.584519i	$-0.801283\pi$
$37$	−9.87086	−1.62276	−0.811380	+	0.584519i	$0.801283\pi$
$38$	0	0
$39$	−8.61676	−1.37979
$40$	0	0
$41$	0.508203	0.0793680	0.0396840	−	0.999212i	$-0.487365\pi$
$41$	0.508203	0.0793680	0.0396840	+	0.999212i	$0.487365\pi$
$42$	0	0
$43$	−1.38324	−0.210942	−0.105471	−	0.994422i	$-0.533635\pi$
$43$	−1.38324	−0.210942	−0.105471	+	0.994422i	$0.533635\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.36266	0.198765	0.0993823	−	0.995049i	$-0.468313\pi$
$47$	1.36266	0.198765	0.0993823	+	0.995049i	$0.468313\pi$
$48$	0	0
$49$	−5.42723	−0.775318
$50$	0	0
$51$	20.9149	2.92867
$52$	0	0
$53$	−2.23769	−0.307371	−0.153686	−	0.988120i	$-0.549114\pi$
$53$	−2.23769	−0.307371	−0.153686	+	0.988120i	$0.549114\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−14.2499	−1.88745
$58$	0	0
$59$	−11.6608	−1.51810	−0.759050	−	0.651032i	$-0.774336\pi$
$59$	−11.6608	−1.51810	−0.759050	+	0.651032i	$0.774336\pi$
$60$	0	0
$61$	3.41082	0.436711	0.218356	−	0.975869i	$-0.429931\pi$
$61$	3.41082	0.436711	0.218356	+	0.975869i	$0.429931\pi$
$62$	0	0
$63$	7.04399	0.887460
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.72532	0.821629	0.410814	−	0.911719i	$-0.365244\pi$
$67$	6.72532	0.821629	0.410814	+	0.911719i	$0.365244\pi$
$68$	0	0
$69$	4.61676	0.555793
$70$	0	0
$71$	14.7253	1.74757	0.873787	−	0.486309i	$-0.161657\pi$
$71$	14.7253	1.74757	0.873787	+	0.486309i	$0.161657\pi$
$72$	0	0
$73$	12.3585	1.44645	0.723226	−	0.690611i	$-0.242658\pi$
$73$	12.3585	1.44645	0.723226	+	0.690611i	$0.242658\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.14554	0.358468
$78$	0	0
$79$	3.91903	0.440925	0.220462	−	0.975395i	$-0.429243\pi$
$79$	3.91903	0.440925	0.220462	+	0.975395i	$0.429243\pi$
$80$	0	0
$81$	5.69774	0.633082
$82$	0	0
$83$	1.87086	0.205354	0.102677	−	0.994715i	$-0.467259\pi$
$83$	1.87086	0.205354	0.102677	+	0.994715i	$0.467259\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.93543	0.314711
$88$	0	0
$89$	−11.8709	−1.25831	−0.629155	−	0.777280i	$-0.716599\pi$
$89$	−11.8709	−1.25831	−0.629155	+	0.777280i	$0.716599\pi$
$90$	0	0
$91$	3.68133	0.385908
$92$	0	0
$93$	13.5522	1.40530
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−5.31450	−0.539606	−0.269803	−	0.962916i	$-0.586959\pi$
$97$	−5.31450	−0.539606	−0.269803	+	0.962916i	$0.586959\pi$
$98$	0	0
$99$	14.0880	1.41590
$100$	0	0
$101$	−13.4436	−1.33769	−0.668846	−	0.743401i	$-0.733211\pi$
$101$	−13.4436	−1.33769	−0.668846	+	0.743401i	$0.733211\pi$
$102$	0	0
$103$	2.03281	0.200299	0.100150	−	0.994972i	$-0.468068\pi$
$103$	2.03281	0.200299	0.100150	+	0.994972i	$0.468068\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.11273	−0.107572	−0.0537858	−	0.998552i	$-0.517129\pi$
$107$	−1.11273	−0.107572	−0.0537858	+	0.998552i	$0.517129\pi$
$108$	0	0
$109$	−6.21712	−0.595492	−0.297746	−	0.954645i	$-0.596235\pi$
$109$	−6.21712	−0.595492	−0.297746	+	0.954645i	$0.596235\pi$
$110$	0	0
$111$	28.9753	2.75021
$112$	0	0
$113$	19.1854	1.80481	0.902404	−	0.430892i	$-0.141801\pi$
$113$	19.1854	1.80481	0.902404	+	0.430892i	$0.141801\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	16.4876	1.52428
$118$	0	0
$119$	−8.93543	−0.819110
$120$	0	0
$121$	−4.70892	−0.428083
$122$	0	0
$123$	−1.49180	−0.134511
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.34625	0.385668	0.192834	−	0.981231i	$-0.438232\pi$
$127$	4.34625	0.385668	0.192834	+	0.981231i	$0.438232\pi$
$128$	0	0
$129$	4.06040	0.357498
$130$	0	0
$131$	−15.1044	−1.31968	−0.659838	−	0.751408i	$-0.729375\pi$
$131$	−15.1044	−1.31968	−0.659838	+	0.751408i	$0.729375\pi$
$132$	0	0
$133$	6.08798	0.527895
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.44364	0.294210	0.147105	−	0.989121i	$-0.453005\pi$
$137$	3.44364	0.294210	0.147105	+	0.989121i	$0.453005\pi$
$138$	0	0
$139$	1.41605	0.120108	0.0600539	−	0.998195i	$-0.480873\pi$
$139$	1.41605	0.120108	0.0600539	+	0.998195i	$0.480873\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	7.36266	0.615697
$144$	0	0
$145$	0	0
$146$	0	0
$147$	15.9313	1.31399
$148$	0	0
$149$	−21.5798	−1.76788	−0.883942	−	0.467596i	$-0.845120\pi$
$149$	−21.5798	−1.76788	−0.883942	+	0.467596i	$0.845120\pi$
$150$	0	0
$151$	18.6290	1.51601	0.758003	−	0.652251i	$-0.226175\pi$
$151$	18.6290	1.51601	0.758003	+	0.652251i	$0.226175\pi$
$152$	0	0
$153$	−40.0192	−3.23536
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−21.0716	−1.68169	−0.840847	−	0.541272i	$-0.817943\pi$
$157$	−21.0716	−1.68169	−0.840847	+	0.541272i	$0.817943\pi$
$158$	0	0
$159$	6.56860	0.520924
$160$	0	0
$161$	−1.97241	−0.155448
$162$	0	0
$163$	−9.36266	−0.733340	−0.366670	−	0.930351i	$-0.619502\pi$
$163$	−9.36266	−0.733340	−0.366670	+	0.930351i	$0.619502\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.2294	−1.10110	−0.550550	−	0.834802i	$-0.685582\pi$
$167$	−14.2294	−1.10110	−0.550550	+	0.834802i	$0.685582\pi$
$168$	0	0
$169$	−4.38324	−0.337172
$170$	0	0
$171$	27.2663	2.08511
$172$	0	0
$173$	−3.95184	−0.300453	−0.150226	−	0.988652i	$-0.548000\pi$
$173$	−3.95184	−0.300453	−0.150226	+	0.988652i	$0.548000\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	34.2294	2.57284
$178$	0	0
$179$	11.6608	0.871566	0.435783	−	0.900052i	$-0.356472\pi$
$179$	11.6608	0.871566	0.435783	+	0.900052i	$0.356472\pi$
$180$	0	0
$181$	11.2335	0.834981	0.417491	−	0.908681i	$-0.362910\pi$
$181$	11.2335	0.834981	0.417491	+	0.908681i	$0.362910\pi$
$182$	0	0
$183$	−10.0122	−0.740126
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−17.8709	−1.30685
$188$	0	0
$189$	−9.63317	−0.700710
$190$	0	0
$191$	15.3421	1.11011	0.555057	−	0.831812i	$-0.312696\pi$
$191$	15.3421	1.11011	0.555057	+	0.831812i	$0.312696\pi$
$192$	0	0
$193$	−2.42723	−0.174716	−0.0873579	−	0.996177i	$-0.527842\pi$
$193$	−2.42723	−0.174716	−0.0873579	+	0.996177i	$0.527842\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.4876	1.17469	0.587347	−	0.809335i	$-0.300172\pi$
$197$	16.4876	1.17469	0.587347	+	0.809335i	$0.300172\pi$
$198$	0	0
$199$	14.2499	1.01015	0.505075	−	0.863075i	$-0.331465\pi$
$199$	14.2499	1.01015	0.505075	+	0.863075i	$0.331465\pi$
$200$	0	0
$201$	−19.7417	−1.39247
$202$	0	0
$203$	−1.25410	−0.0880207
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−8.83388	−0.613997
$208$	0	0
$209$	12.1760	0.842229
$210$	0	0
$211$	15.5246	1.06876	0.534379	−	0.845245i	$-0.320545\pi$
$211$	15.5246	1.06876	0.534379	+	0.845245i	$0.320545\pi$
$212$	0	0
$213$	−43.2252	−2.96174
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.78989	−0.393043
$218$	0	0
$219$	−36.2775	−2.45141
$220$	0	0
$221$	−20.9149	−1.40689
$222$	0	0
$223$	26.1690	1.75240	0.876202	−	0.481945i	$-0.160069\pi$
$223$	26.1690	1.75240	0.876202	+	0.481945i	$0.160069\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.3791	−1.08712	−0.543558	−	0.839371i	$-0.682923\pi$
$227$	−16.3791	−1.08712	−0.543558	+	0.839371i	$0.682923\pi$
$228$	0	0
$229$	−10.1414	−0.670161	−0.335080	−	0.942190i	$-0.608763\pi$
$229$	−10.1414	−0.670161	−0.335080	+	0.942190i	$0.608763\pi$
$230$	0	0
$231$	−9.23353	−0.607521
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−11.5040	−0.747267
$238$	0	0
$239$	−12.3791	−0.800735	−0.400368	−	0.916355i	$-0.631118\pi$
$239$	−12.3791	−0.800735	−0.400368	+	0.916355i	$0.631118\pi$
$240$	0	0
$241$	7.41082	0.477373	0.238687	−	0.971097i	$-0.423283\pi$
$241$	7.41082	0.477373	0.238687	+	0.971097i	$0.423283\pi$
$242$	0	0
$243$	6.31867	0.405343
$244$	0	0
$245$	0	0
$246$	0	0
$247$	14.2499	0.906702
$248$	0	0
$249$	−5.49180	−0.348028
$250$	0	0
$251$	−17.8709	−1.12800	−0.563999	−	0.825775i	$-0.690738\pi$
$251$	−17.8709	−1.12800	−0.563999	+	0.825775i	$0.690738\pi$
$252$	0	0
$253$	−3.94483	−0.248009
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.2499	1.51267	0.756335	−	0.654185i	$-0.226988\pi$
$257$	24.2499	1.51267	0.756335	+	0.654185i	$0.226988\pi$
$258$	0	0
$259$	−12.3791	−0.769198
$260$	0	0
$261$	−5.61676	−0.347669
$262$	0	0
$263$	−8.34625	−0.514652	−0.257326	−	0.966325i	$-0.582841\pi$
$263$	−8.34625	−0.514652	−0.257326	+	0.966325i	$0.582841\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	34.8461	2.13255
$268$	0	0
$269$	−26.8339	−1.63609	−0.818045	−	0.575154i	$-0.804942\pi$
$269$	−26.8339	−1.63609	−0.818045	+	0.575154i	$0.804942\pi$
$270$	0	0
$271$	−25.7745	−1.56569	−0.782846	−	0.622216i	$-0.786233\pi$
$271$	−25.7745	−1.56569	−0.782846	+	0.622216i	$0.786233\pi$
$272$	0	0
$273$	−10.8063	−0.654027
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.6925	0.642451	0.321225	−	0.947003i	$-0.395905\pi$
$277$	10.6925	0.642451	0.321225	+	0.947003i	$0.395905\pi$
$278$	0	0
$279$	−25.9313	−1.55246
$280$	0	0
$281$	−19.3749	−1.15581	−0.577905	−	0.816104i	$-0.696130\pi$
$281$	−19.3749	−1.15581	−0.577905	+	0.816104i	$0.696130\pi$
$282$	0	0
$283$	−1.87086	−0.111211	−0.0556057	−	0.998453i	$-0.517709\pi$
$283$	−1.87086	−0.111211	−0.0556057	+	0.998453i	$0.517709\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.637339	0.0376209
$288$	0	0
$289$	33.7651	1.98619
$290$	0	0
$291$	15.6004	0.914509
$292$	0	0
$293$	4.59619	0.268512	0.134256	−	0.990947i	$-0.457136\pi$
$293$	4.59619	0.268512	0.134256	+	0.990947i	$0.457136\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−19.2663	−1.11795
$298$	0	0
$299$	−4.61676	−0.266994
$300$	0	0
$301$	−1.73472	−0.0999876
$302$	0	0
$303$	39.4629	2.26708
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−8.08798	−0.461606	−0.230803	−	0.973001i	$-0.574135\pi$
$307$	−8.08798	−0.461606	−0.230803	+	0.973001i	$0.574135\pi$
$308$	0	0
$309$	−5.96719	−0.339461
$310$	0	0
$311$	−10.6443	−0.603586	−0.301793	−	0.953374i	$-0.597585\pi$
$311$	−10.6443	−0.603586	−0.301793	+	0.953374i	$0.597585\pi$
$312$	0	0
$313$	−3.96719	−0.224239	−0.112119	−	0.993695i	$-0.535764\pi$
$313$	−3.96719	−0.224239	−0.112119	+	0.993695i	$0.535764\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.2499	−0.800356	−0.400178	−	0.916437i	$-0.631052\pi$
$317$	−14.2499	−0.800356	−0.400178	+	0.916437i	$0.631052\pi$
$318$	0	0
$319$	−2.50820	−0.140432
$320$	0	0
$321$	3.26634	0.182309
$322$	0	0
$323$	−34.5878	−1.92452
$324$	0	0
$325$	0	0
$326$	0	0
$327$	18.2499	1.00922
$328$	0	0
$329$	1.70892	0.0942156
$330$	0	0
$331$	−8.97526	−0.493325	−0.246662	−	0.969101i	$-0.579334\pi$
$331$	−8.97526	−0.493325	−0.246662	+	0.969101i	$0.579334\pi$
$332$	0	0
$333$	−55.4423	−3.03822
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−22.6496	−1.23380	−0.616901	−	0.787041i	$-0.711612\pi$
$337$	−22.6496	−1.23380	−0.616901	+	0.787041i	$0.711612\pi$
$338$	0	0
$339$	−56.3173	−3.05874
$340$	0	0
$341$	−11.5798	−0.627080
$342$	0	0
$343$	−15.5850	−0.841511
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−26.8873	−1.44338	−0.721692	−	0.692214i	$-0.756635\pi$
$347$	−26.8873	−1.44338	−0.721692	+	0.692214i	$0.756635\pi$
$348$	0	0
$349$	22.8133	1.22117	0.610584	−	0.791951i	$-0.290935\pi$
$349$	22.8133	1.22117	0.610584	+	0.791951i	$0.290935\pi$
$350$	0	0
$351$	−22.5480	−1.20352
$352$	0	0
$353$	21.7969	1.16013	0.580066	−	0.814570i	$-0.303027\pi$
$353$	21.7969	1.16013	0.580066	+	0.814570i	$0.303027\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	26.2294	1.38820
$358$	0	0
$359$	−17.9518	−0.947462	−0.473731	−	0.880670i	$-0.657093\pi$
$359$	−17.9518	−0.947462	−0.473731	+	0.880670i	$0.657093\pi$
$360$	0	0
$361$	4.56576	0.240303
$362$	0	0
$363$	13.8227	0.725504
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−19.9917	−1.04356	−0.521778	−	0.853081i	$-0.674731\pi$
$367$	−19.9917	−1.04356	−0.521778	+	0.853081i	$0.674731\pi$
$368$	0	0
$369$	2.85446	0.148597
$370$	0	0
$371$	−2.80630	−0.145696
$372$	0	0
$373$	−27.6883	−1.43365	−0.716824	−	0.697254i	$-0.754405\pi$
$373$	−27.6883	−1.43365	−0.716824	+	0.697254i	$0.754405\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.93543	−0.151182
$378$	0	0
$379$	−36.7170	−1.88602	−0.943012	−	0.332758i	$-0.892021\pi$
$379$	−36.7170	−1.88602	−0.943012	+	0.332758i	$0.892021\pi$
$380$	0	0
$381$	−12.7581	−0.653619
$382$	0	0
$383$	−10.2294	−0.522696	−0.261348	−	0.965245i	$-0.584167\pi$
$383$	−10.2294	−0.522696	−0.261348	+	0.965245i	$0.584167\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.76931	−0.394936
$388$	0	0
$389$	−2.75814	−0.139843	−0.0699215	−	0.997552i	$-0.522275\pi$
$389$	−2.75814	−0.139843	−0.0699215	+	0.997552i	$0.522275\pi$
$390$	0	0
$391$	11.2059	0.566709
$392$	0	0
$393$	44.3379	2.23655
$394$	0	0
$395$	0	0
$396$	0	0
$397$	15.6883	0.787375	0.393688	−	0.919244i	$-0.371199\pi$
$397$	15.6883	0.787375	0.393688	+	0.919244i	$0.371199\pi$
$398$	0	0
$399$	−17.8709	−0.894662
$400$	0	0
$401$	−31.3145	−1.56377	−0.781886	−	0.623422i	$-0.785742\pi$
$401$	−31.3145	−1.56377	−0.781886	+	0.623422i	$0.785742\pi$
$402$	0	0
$403$	−13.5522	−0.675083
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−24.7581	−1.22722
$408$	0	0
$409$	−17.8625	−0.883245	−0.441623	−	0.897201i	$-0.645597\pi$
$409$	−17.8625	−0.883245	−0.441623	+	0.897201i	$0.645597\pi$
$410$	0	0
$411$	−10.1086	−0.498618
$412$	0	0
$413$	−14.6238	−0.719589
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.15672	−0.203555
$418$	0	0
$419$	38.0674	1.85971	0.929857	−	0.367920i	$-0.119930\pi$
$419$	38.0674	1.85971	0.929857	+	0.367920i	$0.119930\pi$
$420$	0	0
$421$	−26.4999	−1.29152	−0.645762	−	0.763539i	$-0.723460\pi$
$421$	−26.4999	−1.29152	−0.645762	+	0.763539i	$0.723460\pi$
$422$	0	0
$423$	7.65375	0.372138
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.27752	0.207004
$428$	0	0
$429$	−21.6126	−1.04347
$430$	0	0
$431$	−30.3707	−1.46291	−0.731453	−	0.681892i	$-0.761158\pi$
$431$	−30.3707	−1.46291	−0.731453	+	0.681892i	$0.761158\pi$
$432$	0	0
$433$	−8.63734	−0.415084	−0.207542	−	0.978226i	$-0.566546\pi$
$433$	−8.63734	−0.415084	−0.207542	+	0.978226i	$0.566546\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.63495	−0.365229
$438$	0	0
$439$	4.70297	0.224460	0.112230	−	0.993682i	$-0.464201\pi$
$439$	4.70297	0.224460	0.112230	+	0.993682i	$0.464201\pi$
$440$	0	0
$441$	−30.4835	−1.45159
$442$	0	0
$443$	3.25933	0.154855	0.0774277	−	0.996998i	$-0.475329\pi$
$443$	3.25933	0.154855	0.0774277	+	0.996998i	$0.475329\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	63.3460	2.99616
$448$	0	0
$449$	36.7498	1.73433	0.867165	−	0.498021i	$-0.165940\pi$
$449$	36.7498	1.73433	0.867165	+	0.498021i	$0.165940\pi$
$450$	0	0
$451$	1.27468	0.0600222
$452$	0	0
$453$	−54.6842	−2.56929
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.9508	−0.512256	−0.256128	−	0.966643i	$-0.582447\pi$
$457$	−10.9508	−0.512256	−0.256128	+	0.966643i	$0.582447\pi$
$458$	0	0
$459$	54.7292	2.55454
$460$	0	0
$461$	26.4241	1.23069	0.615347	−	0.788256i	$-0.289016\pi$
$461$	26.4241	1.23069	0.615347	+	0.788256i	$0.289016\pi$
$462$	0	0
$463$	29.5163	1.37174	0.685869	−	0.727725i	$-0.259422\pi$
$463$	29.5163	1.37174	0.685869	+	0.727725i	$0.259422\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.38324	−0.249107	−0.124553	−	0.992213i	$-0.539750\pi$
$467$	−5.38324	−0.249107	−0.124553	+	0.992213i	$0.539750\pi$
$468$	0	0
$469$	8.43424	0.389457
$470$	0	0
$471$	61.8542	2.85009
$472$	0	0
$473$	−3.46944	−0.159525
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−12.5686	−0.575477
$478$	0	0
$479$	−14.6084	−0.667476	−0.333738	−	0.942666i	$-0.608310\pi$
$479$	−14.6084	−0.667476	−0.333738	+	0.942666i	$0.608310\pi$
$480$	0	0
$481$	−28.9753	−1.32116
$482$	0	0
$483$	5.78989	0.263449
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5.41082	−0.245188	−0.122594	−	0.992457i	$-0.539121\pi$
$487$	−5.41082	−0.245188	−0.122594	+	0.992457i	$0.539121\pi$
$488$	0	0
$489$	27.4835	1.24284
$490$	0	0
$491$	16.6925	0.753322	0.376661	−	0.926351i	$-0.377072\pi$
$491$	16.6925	0.753322	0.376661	+	0.926351i	$0.377072\pi$
$492$	0	0
$493$	7.12497	0.320892
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.4671	0.828360
$498$	0	0
$499$	30.8859	1.38264	0.691322	−	0.722546i	$-0.257028\pi$
$499$	30.8859	1.38264	0.691322	+	0.722546i	$0.257028\pi$
$500$	0	0
$501$	41.7693	1.86612
$502$	0	0
$503$	21.0388	0.938072	0.469036	−	0.883179i	$-0.344601\pi$
$503$	21.0388	0.938072	0.469036	+	0.883179i	$0.344601\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.8667	0.571430
$508$	0	0
$509$	−37.7006	−1.67105	−0.835524	−	0.549453i	$-0.814836\pi$
$509$	−37.7006	−1.67105	−0.835524	+	0.549453i	$0.814836\pi$
$510$	0	0
$511$	15.4988	0.685627
$512$	0	0
$513$	−37.2887	−1.64634
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.41783	0.150316
$518$	0	0
$519$	11.6004	0.509199
$520$	0	0
$521$	27.5920	1.20883	0.604414	−	0.796670i	$-0.293407\pi$
$521$	27.5920	1.20883	0.604414	+	0.796670i	$0.293407\pi$
$522$	0	0
$523$	−40.6618	−1.77802	−0.889008	−	0.457891i	$-0.848605\pi$
$523$	−40.6618	−1.77802	−0.889008	+	0.457891i	$0.848605\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	32.8943	1.43290
$528$	0	0
$529$	−20.5264	−0.892452
$530$	0	0
$531$	−65.4957	−2.84227
$532$	0	0
$533$	1.49180	0.0646169
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−34.2294	−1.47711
$538$	0	0
$539$	−13.6126	−0.586336
$540$	0	0
$541$	26.4241	1.13606	0.568031	−	0.823007i	$-0.307705\pi$
$541$	26.4241	1.13606	0.568031	+	0.823007i	$0.307705\pi$
$542$	0	0
$543$	−32.9753	−1.41510
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.54697	0.0661435	0.0330718	−	0.999453i	$-0.489471\pi$
$547$	1.54697	0.0661435	0.0330718	+	0.999453i	$0.489471\pi$
$548$	0	0
$549$	19.1578	0.817634
$550$	0	0
$551$	−4.85446	−0.206807
$552$	0	0
$553$	4.91486	0.209001
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.09738	0.300726	0.150363	−	0.988631i	$-0.451956\pi$
$557$	7.09738	0.300726	0.150363	+	0.988631i	$0.451956\pi$
$558$	0	0
$559$	−4.06040	−0.171737
$560$	0	0
$561$	52.4587	2.21481
$562$	0	0
$563$	−3.25933	−0.137364	−0.0686822	−	0.997639i	$-0.521879\pi$
$563$	−3.25933	−0.137364	−0.0686822	+	0.997639i	$0.521879\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	7.14554	0.300085
$568$	0	0
$569$	−2.81331	−0.117940	−0.0589700	−	0.998260i	$-0.518782\pi$
$569$	−2.81331	−0.117940	−0.0589700	+	0.998260i	$0.518782\pi$
$570$	0	0
$571$	35.3369	1.47880	0.739401	−	0.673266i	$-0.235109\pi$
$571$	35.3369	1.47880	0.739401	+	0.673266i	$0.235109\pi$
$572$	0	0
$573$	−45.0357	−1.88139
$574$	0	0
$575$	0	0
$576$	0	0
$577$	21.6332	0.900601	0.450300	−	0.892877i	$-0.351317\pi$
$577$	21.6332	0.900601	0.450300	+	0.892877i	$0.351317\pi$
$578$	0	0
$579$	7.12497	0.296103
$580$	0	0
$581$	2.34625	0.0973390
$582$	0	0
$583$	−5.61259	−0.232450
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−44.1208	−1.82106	−0.910530	−	0.413443i	$-0.864326\pi$
$587$	−44.1208	−1.82106	−0.910530	+	0.413443i	$0.864326\pi$
$588$	0	0
$589$	−22.4119	−0.923465
$590$	0	0
$591$	−48.3983	−1.99084
$592$	0	0
$593$	17.3871	0.714004	0.357002	−	0.934104i	$-0.383799\pi$
$593$	17.3871	0.714004	0.357002	+	0.934104i	$0.383799\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−41.8297	−1.71198
$598$	0	0
$599$	17.5920	0.718790	0.359395	−	0.933185i	$-0.382983\pi$
$599$	17.5920	0.718790	0.359395	+	0.933185i	$0.382983\pi$
$600$	0	0
$601$	6.78883	0.276922	0.138461	−	0.990368i	$-0.455784\pi$
$601$	6.78883	0.276922	0.138461	+	0.990368i	$0.455784\pi$
$602$	0	0
$603$	37.7745	1.53830
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.60453	0.349247	0.174623	−	0.984635i	$-0.444129\pi$
$607$	8.60453	0.349247	0.174623	+	0.984635i	$0.444129\pi$
$608$	0	0
$609$	3.68133	0.149175
$610$	0	0
$611$	4.00000	0.161823
$612$	0	0
$613$	−38.4137	−1.55151	−0.775757	−	0.631032i	$-0.782632\pi$
$613$	−38.4137	−1.55151	−0.775757	+	0.631032i	$0.782632\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.15778	0.368678	0.184339	−	0.982863i	$-0.440986\pi$
$617$	9.15778	0.368678	0.184339	+	0.982863i	$0.440986\pi$
$618$	0	0
$619$	−30.1536	−1.21198	−0.605988	−	0.795474i	$-0.707222\pi$
$619$	−30.1536	−1.21198	−0.605988	+	0.795474i	$0.707222\pi$
$620$	0	0
$621$	12.0810	0.484793
$622$	0	0
$623$	−14.8873	−0.596446
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−35.7417	−1.42739
$628$	0	0
$629$	70.3296	2.80422
$630$	0	0
$631$	21.6537	0.862022	0.431011	−	0.902347i	$-0.358157\pi$
$631$	21.6537	0.862022	0.431011	+	0.902347i	$0.358157\pi$
$632$	0	0
$633$	−45.5714	−1.81130
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−15.9313	−0.631220
$638$	0	0
$639$	82.7086	3.27190
$640$	0	0
$641$	−49.4423	−1.95285	−0.976427	−	0.215849i	$-0.930748\pi$
$641$	−49.4423	−1.95285	−0.976427	+	0.215849i	$0.930748\pi$
$642$	0	0
$643$	−23.4423	−0.924474	−0.462237	−	0.886756i	$-0.652953\pi$
$643$	−23.4423	−0.924474	−0.462237	+	0.886756i	$0.652953\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.2911	0.561840	0.280920	−	0.959731i	$-0.409360\pi$
$647$	14.2911	0.561840	0.280920	+	0.959731i	$0.409360\pi$
$648$	0	0
$649$	−29.2475	−1.14807
$650$	0	0
$651$	16.9958	0.666119
$652$	0	0
$653$	−23.5246	−0.920589	−0.460295	−	0.887766i	$-0.652256\pi$
$653$	−23.5246	−0.920589	−0.460295	+	0.887766i	$0.652256\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	69.4147	2.70813
$658$	0	0
$659$	−6.24993	−0.243463	−0.121731	−	0.992563i	$-0.538845\pi$
$659$	−6.24993	−0.243463	−0.121731	+	0.992563i	$0.538845\pi$
$660$	0	0
$661$	20.6290	0.802375	0.401188	−	0.915996i	$-0.368598\pi$
$661$	20.6290	0.802375	0.401188	+	0.915996i	$0.368598\pi$
$662$	0	0
$663$	61.3941	2.38435
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.57277	0.0608979
$668$	0	0
$669$	−76.8172	−2.96992
$670$	0	0
$671$	8.55504	0.330263
$672$	0	0
$673$	27.2335	1.04978	0.524888	−	0.851172i	$-0.324107\pi$
$673$	27.2335	1.04978	0.524888	+	0.851172i	$0.324107\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−24.8238	−0.954055	−0.477027	−	0.878888i	$-0.658286\pi$
$677$	−24.8238	−0.954055	−0.477027	+	0.878888i	$0.658286\pi$
$678$	0	0
$679$	−6.66492	−0.255776
$680$	0	0
$681$	48.0796	1.84242
$682$	0	0
$683$	34.3707	1.31516	0.657580	−	0.753385i	$-0.271580\pi$
$683$	34.3707	1.31516	0.657580	+	0.753385i	$0.271580\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	29.7693	1.13577
$688$	0	0
$689$	−6.56860	−0.250244
$690$	0	0
$691$	17.0922	0.650216	0.325108	−	0.945677i	$-0.394599\pi$
$691$	17.0922	0.650216	0.325108	+	0.945677i	$0.394599\pi$
$692$	0	0
$693$	17.6678	0.671143
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−3.62093	−0.137153
$698$	0	0
$699$	52.8378	1.99851
$700$	0	0
$701$	24.8050	0.936871	0.468435	−	0.883498i	$-0.344818\pi$
$701$	24.8050	0.936871	0.468435	+	0.883498i	$0.344818\pi$
$702$	0	0
$703$	−47.9177	−1.80725
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16.8597	−0.634074
$708$	0	0
$709$	−41.5631	−1.56094	−0.780468	−	0.625196i	$-0.785019\pi$
$709$	−41.5631	−1.56094	−0.780468	+	0.625196i	$0.785019\pi$
$710$	0	0
$711$	22.0122	0.825523
$712$	0	0
$713$	7.26111	0.271931
$714$	0	0
$715$	0	0
$716$	0	0
$717$	36.3379	1.35706
$718$	0	0
$719$	−10.6290	−0.396395	−0.198197	−	0.980162i	$-0.563509\pi$
$719$	−10.6290	−0.396395	−0.198197	+	0.980162i	$0.563509\pi$
$720$	0	0
$721$	2.54935	0.0949429
$722$	0	0
$723$	−21.7540	−0.809039
$724$	0	0
$725$	0	0
$726$	0	0
$727$	10.8545	0.402570	0.201285	−	0.979533i	$-0.435488\pi$
$727$	10.8545	0.402570	0.201285	+	0.979533i	$0.435488\pi$
$728$	0	0
$729$	−35.6412	−1.32005
$730$	0	0
$731$	9.85552	0.364519
$732$	0	0
$733$	−11.0492	−0.408112	−0.204056	−	0.978959i	$-0.565413\pi$
$733$	−11.0492	−0.408112	−0.204056	+	0.978959i	$0.565413\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.8685	0.621358
$738$	0	0
$739$	40.7581	1.49931	0.749656	−	0.661828i	$-0.230219\pi$
$739$	40.7581	1.49931	0.749656	+	0.661828i	$0.230219\pi$
$740$	0	0
$741$	−41.8297	−1.53665
$742$	0	0
$743$	−8.08798	−0.296719	−0.148360	−	0.988933i	$-0.547399\pi$
$743$	−8.08798	−0.296719	−0.148360	+	0.988933i	$0.547399\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10.5082	0.384475
$748$	0	0
$749$	−1.39547	−0.0509895
$750$	0	0
$751$	−6.20878	−0.226562	−0.113281	−	0.993563i	$-0.536136\pi$
$751$	−6.20878	−0.226562	−0.113281	+	0.993563i	$0.536136\pi$
$752$	0	0
$753$	52.4587	1.91170
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−20.0963	−0.730413	−0.365207	−	0.930926i	$-0.619002\pi$
$757$	−20.0963	−0.730413	−0.365207	+	0.930926i	$0.619002\pi$
$758$	0	0
$759$	11.5798	0.420319
$760$	0	0
$761$	13.0838	0.474288	0.237144	−	0.971475i	$-0.423789\pi$
$761$	13.0838	0.474288	0.237144	+	0.971475i	$0.423789\pi$
$762$	0	0
$763$	−7.79690	−0.282267
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−34.2294	−1.23595
$768$	0	0
$769$	3.61259	0.130273	0.0651367	−	0.997876i	$-0.479252\pi$
$769$	3.61259	0.130273	0.0651367	+	0.997876i	$0.479252\pi$
$770$	0	0
$771$	−71.1840	−2.56363
$772$	0	0
$773$	38.0880	1.36993	0.684965	−	0.728576i	$-0.259818\pi$
$773$	38.0880	1.36993	0.684965	+	0.728576i	$0.259818\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	36.3379	1.30362
$778$	0	0
$779$	2.46705	0.0883913
$780$	0	0
$781$	36.9341	1.32161
$782$	0	0
$783$	7.68133	0.274508
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−54.5467	−1.94438	−0.972190	−	0.234194i	$-0.924755\pi$
$787$	−54.5467	−1.94438	−0.972190	+	0.234194i	$0.924755\pi$
$788$	0	0
$789$	24.4999	0.872218
$790$	0	0
$791$	24.0604	0.855489
$792$	0	0
$793$	10.0122	0.355545
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.27468	−0.0451514	−0.0225757	−	0.999745i	$-0.507187\pi$
$797$	−1.27468	−0.0451514	−0.0225757	+	0.999745i	$0.507187\pi$
$798$	0	0
$799$	−9.70892	−0.343477
$800$	0	0
$801$	−66.6758	−2.35587
$802$	0	0
$803$	30.9976	1.09388
$804$	0	0
$805$	0	0
$806$	0	0
$807$	78.7690	2.77280
$808$	0	0
$809$	21.3627	0.751071	0.375536	−	0.926808i	$-0.377459\pi$
$809$	21.3627	0.751071	0.375536	+	0.926808i	$0.377459\pi$
$810$	0	0
$811$	−23.8175	−0.836345	−0.418172	−	0.908368i	$-0.637329\pi$
$811$	−23.8175	−0.836345	−0.418172	+	0.908368i	$0.637329\pi$
$812$	0	0
$813$	75.6594	2.65349
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−6.71487	−0.234923
$818$	0	0
$819$	20.6772	0.722519
$820$	0	0
$821$	6.59619	0.230209	0.115104	−	0.993353i	$-0.463280\pi$
$821$	6.59619	0.230209	0.115104	+	0.993353i	$0.463280\pi$
$822$	0	0
$823$	2.00000	0.0697156	0.0348578	−	0.999392i	$-0.488902\pi$
$823$	2.00000	0.0697156	0.0348578	+	0.999392i	$0.488902\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18.5511	−0.645086	−0.322543	−	0.946555i	$-0.604538\pi$
$827$	−18.5511	−0.645086	−0.322543	+	0.946555i	$0.604538\pi$
$828$	0	0
$829$	6.55114	0.227530	0.113765	−	0.993508i	$-0.463709\pi$
$829$	6.55114	0.227530	0.113765	+	0.993508i	$0.463709\pi$
$830$	0	0
$831$	−31.3871	−1.08881
$832$	0	0
$833$	38.6688	1.33980
$834$	0	0
$835$	0	0
$836$	0	0
$837$	35.4629	1.22578
$838$	0	0
$839$	−43.9833	−1.51847	−0.759236	−	0.650815i	$-0.774427\pi$
$839$	−43.9833	−1.51847	−0.759236	+	0.650815i	$0.774427\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	56.8737	1.95883
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5.90546	−0.202914
$848$	0	0
$849$	5.49180	0.188478
$850$	0	0
$851$	15.5246	0.532177
$852$	0	0
$853$	−16.8545	−0.577086	−0.288543	−	0.957467i	$-0.593171\pi$
$853$	−16.8545	−0.577086	−0.288543	+	0.957467i	$0.593171\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−14.0796	−0.480952	−0.240476	−	0.970655i	$-0.577303\pi$
$857$	−14.0796	−0.480952	−0.240476	+	0.970655i	$0.577303\pi$
$858$	0	0
$859$	−48.5795	−1.65751	−0.828756	−	0.559610i	$-0.810951\pi$
$859$	−48.5795	−1.65751	−0.828756	+	0.559610i	$0.810951\pi$
$860$	0	0
$861$	−1.87086	−0.0637589
$862$	0	0
$863$	−8.13614	−0.276958	−0.138479	−	0.990365i	$-0.544221\pi$
$863$	−8.13614	−0.276958	−0.138479	+	0.990365i	$0.544221\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−99.1153	−3.36613
$868$	0	0
$869$	9.82971	0.333450
$870$	0	0
$871$	19.7417	0.668923
$872$	0	0
$873$	−29.8503	−1.01028
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−37.8367	−1.27766	−0.638828	−	0.769350i	$-0.720580\pi$
$877$	−37.8367	−1.27766	−0.638828	+	0.769350i	$0.720580\pi$
$878$	0	0
$879$	−13.4918	−0.455067
$880$	0	0
$881$	−36.5082	−1.22999	−0.614996	−	0.788530i	$-0.710843\pi$
$881$	−36.5082	−1.22999	−0.614996	+	0.788530i	$0.710843\pi$
$882$	0	0
$883$	18.8050	0.632838	0.316419	−	0.948620i	$-0.397520\pi$
$883$	18.8050	0.632838	0.316419	+	0.948620i	$0.397520\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39.6371	−1.33088	−0.665441	−	0.746450i	$-0.731757\pi$
$887$	−39.6371	−1.33088	−0.665441	+	0.746450i	$0.731757\pi$
$888$	0	0
$889$	5.45065	0.182809
$890$	0	0
$891$	14.2911	0.478769
$892$	0	0
$893$	6.61498	0.221362
$894$	0	0
$895$	0	0
$896$	0	0
$897$	13.5522	0.452495
$898$	0	0
$899$	4.61676	0.153978
$900$	0	0
$901$	15.9435	0.531155
$902$	0	0
$903$	5.09215	0.169456
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.38324	0.0459296	0.0229648	−	0.999736i	$-0.492689\pi$
$907$	1.38324	0.0459296	0.0229648	+	0.999736i	$0.492689\pi$
$908$	0	0
$909$	−75.5097	−2.50450
$910$	0	0
$911$	−23.3369	−0.773185	−0.386592	−	0.922251i	$-0.626348\pi$
$911$	−23.3369	−0.773185	−0.386592	+	0.922251i	$0.626348\pi$
$912$	0	0
$913$	4.69251	0.155299
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−18.9424	−0.625535
$918$	0	0
$919$	46.2088	1.52429	0.762144	−	0.647408i	$-0.224147\pi$
$919$	46.2088	1.52429	0.762144	+	0.647408i	$0.224147\pi$
$920$	0	0
$921$	23.7417	0.782317
$922$	0	0
$923$	43.2252	1.42277
$924$	0	0
$925$	0	0
$926$	0	0
$927$	11.4178	0.375011
$928$	0	0
$929$	25.4077	0.833600	0.416800	−	0.908998i	$-0.363152\pi$
$929$	25.4077	0.833600	0.416800	+	0.908998i	$0.363152\pi$
$930$	0	0
$931$	−26.3463	−0.863464
$932$	0	0
$933$	31.2458	1.02294
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−36.4259	−1.18998	−0.594991	−	0.803732i	$-0.702844\pi$
$937$	−36.4259	−1.18998	−0.594991	+	0.803732i	$0.702844\pi$
$938$	0	0
$939$	11.6454	0.380034
$940$	0	0
$941$	−41.3627	−1.34838	−0.674192	−	0.738556i	$-0.735508\pi$
$941$	−41.3627	−1.34838	−0.674192	+	0.738556i	$0.735508\pi$
$942$	0	0
$943$	−0.799288	−0.0260284
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−25.6608	−0.833862	−0.416931	−	0.908938i	$-0.636894\pi$
$947$	−25.6608	−0.833862	−0.416931	+	0.908938i	$0.636894\pi$
$948$	0	0
$949$	36.2775	1.17762
$950$	0	0
$951$	41.8297	1.35642
$952$	0	0
$953$	55.1924	1.78786	0.893928	−	0.448210i	$-0.147938\pi$
$953$	55.1924	1.78786	0.893928	+	0.448210i	$0.147938\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.36266	0.238001
$958$	0	0
$959$	4.31867	0.139457
$960$	0	0
$961$	−9.68550	−0.312435
$962$	0	0
$963$	−6.24993	−0.201401
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−8.04683	−0.258769	−0.129384	−	0.991595i	$-0.541300\pi$
$967$	−8.04683	−0.258769	−0.129384	+	0.991595i	$0.541300\pi$
$968$	0	0
$969$	101.530	3.26162
$970$	0	0
$971$	−16.6618	−0.534703	−0.267352	−	0.963599i	$-0.586148\pi$
$971$	−16.6618	−0.534703	−0.267352	+	0.963599i	$0.586148\pi$
$972$	0	0
$973$	1.77587	0.0569318
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.9037	0.956703	0.478352	−	0.878168i	$-0.341234\pi$
$977$	29.9037	0.956703	0.478352	+	0.878168i	$0.341234\pi$
$978$	0	0
$979$	−29.7745	−0.951599
$980$	0	0
$981$	−34.9201	−1.11491
$982$	0	0
$983$	−42.8685	−1.36729	−0.683646	−	0.729814i	$-0.739607\pi$
$983$	−42.8685	−1.36729	−0.683646	+	0.729814i	$0.739607\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5.01641	−0.159674
$988$	0	0
$989$	2.17552	0.0691774
$990$	0	0
$991$	−50.4259	−1.60183	−0.800916	−	0.598777i	$-0.795654\pi$
$991$	−50.4259	−1.60183	−0.800916	+	0.598777i	$0.795654\pi$
$992$	0	0
$993$	26.3463	0.836073
$994$	0	0
$995$	0	0
$996$	0	0
$997$	29.9260	0.947767	0.473884	−	0.880588i	$-0.342852\pi$
$997$	29.9260	0.947767	0.473884	+	0.880588i	$0.342852\pi$
$998$	0	0
$999$	75.8214	2.39888

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5800.2.a.q.1.1		3
5.4	even	2		1160.2.a.g.1.3	✓	3
20.19	odd	2		2320.2.a.p.1.1		3
40.19	odd	2		9280.2.a.bq.1.3		3
40.29	even	2		9280.2.a.bo.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.2.a.g.1.3	✓	3	5.4	even	2
2320.2.a.p.1.1		3	20.19	odd	2
5800.2.a.q.1.1		3	1.1	even	1	trivial
9280.2.a.bo.1.1		3	40.29	even	2
9280.2.a.bq.1.3		3	40.19	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:	\( N \)	\(=\)	\( 5800 = 2^{3} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5800.a (trivial)

\(f(q)\)	\(=\)	\(q-2.93543 q^{3} +1.25410 q^{7} +5.61676 q^{9} +2.50820 q^{11} +2.93543 q^{13} -7.12497 q^{17} +4.85446 q^{19} -3.68133 q^{21} -1.57277 q^{23} -7.68133 q^{27} -1.00000 q^{29} -4.61676 q^{31} -7.36266 q^{33} -9.87086 q^{37} -8.61676 q^{39} +0.508203 q^{41} -1.38324 q^{43} +1.36266 q^{47} -5.42723 q^{49} +20.9149 q^{51} -2.23769 q^{53} -14.2499 q^{57} -11.6608 q^{59} +3.41082 q^{61} +7.04399 q^{63} +6.72532 q^{67} +4.61676 q^{69} +14.7253 q^{71} +12.3585 q^{73} +3.14554 q^{77} +3.91903 q^{79} +5.69774 q^{81} +1.87086 q^{83} +2.93543 q^{87} -11.8709 q^{89} +3.68133 q^{91} +13.5522 q^{93} -5.31450 q^{97} +14.0880 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 3 q^{7} + 2 q^{9} + 6 q^{11} + q^{13} - 5 q^{17} + 2 q^{19} - 4 q^{21} - 11 q^{23} - 16 q^{27} - 3 q^{29} + q^{31} - 8 q^{33} - 14 q^{37} - 11 q^{39} - 19 q^{43} - 10 q^{47} - 10 q^{49}+ \cdots + 9 q^{97}+O(q^{100}) \) 3 * q - q^3 + 3 * q^7 + 2 * q^9 + 6 * q^11 + q^13 - 5 * q^17 + 2 * q^19 - 4 * q^21 - 11 * q^23 - 16 * q^27 - 3 * q^29 + q^31 - 8 * q^33 - 14 * q^37 - 11 * q^39 - 19 * q^43 - 10 * q^47 - 10 * q^49 + 26 * q^51 - 9 * q^53 - 10 * q^57 + q^59 + 7 * q^61 - 8 * q^67 - q^69 + 16 * q^71 - 9 * q^73 + 22 * q^77 + 7 * q^79 + 7 * q^81 - 10 * q^83 + q^87 - 20 * q^89 + 4 * q^91 + 18 * q^93 + 9 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more