LMFDB - Embedded newform 7500.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, 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("7500.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.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:	$59.8878015160$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6x^{2} + 7x + 11 $ x^4 - 2x^3 - 6x^2 + 7*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.12233$ of defining polynomial
Character	$\chi$	$=$	7500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.50430 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.50430 q^{7} +1.00000 q^{9} -6.17438 q^{11} -3.55634 q^{13} +0.495700 q^{17} +0.311674 q^{19} -1.50430 q^{21} -3.06064 q^{23} -1.00000 q^{27} -0.122334 q^{29} -2.94362 q^{31} +6.17438 q^{33} -4.36700 q^{37} +3.55634 q^{39} +4.25327 q^{41} -3.62663 q^{43} +5.28215 q^{47} -4.73708 q^{49} -0.495700 q^{51} -8.59978 q^{53} -0.311674 q^{57} +12.8826 q^{59} -11.3209 q^{61} +1.50430 q^{63} +13.2414 q^{67} +3.06064 q^{69} +3.26823 q^{71} +15.9668 q^{73} -9.28811 q^{77} -8.52746 q^{79} +1.00000 q^{81} +11.3381 q^{83} +0.122334 q^{87} +14.1326 q^{89} -5.34980 q^{91} +2.94362 q^{93} -11.4828 q^{97} -6.17438 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9} - q^{11} + 5 q^{13} + 4 q^{17} - 5 q^{19} - 4 q^{21} + 9 q^{23} - 4 q^{27} + 6 q^{29} + 11 q^{31} + q^{33} + 2 q^{37} - 5 q^{39} - 6 q^{43} + 16 q^{47} - 4 q^{49} - 4 q^{51} - 2 q^{53} + 5 q^{57} + q^{59} - 22 q^{61} + 4 q^{63} + 36 q^{67} - 9 q^{69} + 20 q^{71} + 12 q^{73} - 11 q^{77} - 3 q^{79} + 4 q^{81} + 14 q^{83} - 6 q^{87} - 15 q^{89} - 10 q^{91} - 11 q^{93} - 12 q^{97} - q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^7 + 4 * q^9 - q^11 + 5 * q^13 + 4 * q^17 - 5 * q^19 - 4 * q^21 + 9 * q^23 - 4 * q^27 + 6 * q^29 + 11 * q^31 + q^33 + 2 * q^37 - 5 * q^39 - 6 * q^43 + 16 * q^47 - 4 * q^49 - 4 * q^51 - 2 * q^53 + 5 * q^57 + q^59 - 22 * q^61 + 4 * q^63 + 36 * q^67 - 9 * q^69 + 20 * q^71 + 12 * q^73 - 11 * q^77 - 3 * q^79 + 4 * q^81 + 14 * q^83 - 6 * q^87 - 15 * q^89 - 10 * q^91 - 11 * q^93 - 12 * q^97 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.50430	0.568572	0.284286	−	0.958740i	$-0.408244\pi$
$7$	1.50430	0.568572	0.284286	+	0.958740i	$0.408244\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.17438	−1.86164	−0.930822	−	0.365473i	$-0.880907\pi$
$11$	−6.17438	−1.86164	−0.930822	+	0.365473i	$0.880907\pi$
$12$	0	0
$13$	−3.55634	−0.986352	−0.493176	−	0.869930i	$-0.664164\pi$
$13$	−3.55634	−0.986352	−0.493176	+	0.869930i	$0.664164\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.495700	0.120225	0.0601125	−	0.998192i	$-0.480854\pi$
$17$	0.495700	0.120225	0.0601125	+	0.998192i	$0.480854\pi$
$18$	0	0
$19$	0.311674	0.0715030	0.0357515	−	0.999361i	$-0.488618\pi$
$19$	0.311674	0.0715030	0.0357515	+	0.999361i	$0.488618\pi$
$20$	0	0
$21$	−1.50430	−0.328265
$22$	0	0
$23$	−3.06064	−0.638188	−0.319094	−	0.947723i	$-0.603379\pi$
$23$	−3.06064	−0.638188	−0.319094	+	0.947723i	$0.603379\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.122334	−0.0227168	−0.0113584	−	0.999935i	$-0.503616\pi$
$29$	−0.122334	−0.0227168	−0.0113584	+	0.999935i	$0.503616\pi$
$30$	0	0
$31$	−2.94362	−0.528690	−0.264345	−	0.964428i	$-0.585156\pi$
$31$	−2.94362	−0.528690	−0.264345	+	0.964428i	$0.585156\pi$
$32$	0	0
$33$	6.17438	1.07482
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.36700	−0.717930	−0.358965	−	0.933351i	$-0.616870\pi$
$37$	−4.36700	−0.717930	−0.358965	+	0.933351i	$0.616870\pi$
$38$	0	0
$39$	3.55634	0.569470
$40$	0	0
$41$	4.25327	0.664249	0.332124	−	0.943236i	$-0.392235\pi$
$41$	4.25327	0.664249	0.332124	+	0.943236i	$0.392235\pi$
$42$	0	0
$43$	−3.62663	−0.553056	−0.276528	−	0.961006i	$-0.589184\pi$
$43$	−3.62663	−0.553056	−0.276528	+	0.961006i	$0.589184\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.28215	0.770480	0.385240	−	0.922816i	$-0.374119\pi$
$47$	5.28215	0.770480	0.385240	+	0.922816i	$0.374119\pi$
$48$	0	0
$49$	−4.73708	−0.676726
$50$	0	0
$51$	−0.495700	−0.0694120
$52$	0	0
$53$	−8.59978	−1.18127	−0.590636	−	0.806938i	$-0.701123\pi$
$53$	−8.59978	−1.18127	−0.590636	+	0.806938i	$0.701123\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.311674	−0.0412823
$58$	0	0
$59$	12.8826	1.67717	0.838584	−	0.544772i	$-0.183384\pi$
$59$	12.8826	1.67717	0.838584	+	0.544772i	$0.183384\pi$
$60$	0	0
$61$	−11.3209	−1.44950	−0.724748	−	0.689014i	$-0.758044\pi$
$61$	−11.3209	−1.44950	−0.724748	+	0.689014i	$0.758044\pi$
$62$	0	0
$63$	1.50430	0.189524
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.2414	1.61769	0.808846	−	0.588020i	$-0.200092\pi$
$67$	13.2414	1.61769	0.808846	+	0.588020i	$0.200092\pi$
$68$	0	0
$69$	3.06064	0.368458
$70$	0	0
$71$	3.26823	0.387868	0.193934	−	0.981015i	$-0.437875\pi$
$71$	3.26823	0.387868	0.193934	+	0.981015i	$0.437875\pi$
$72$	0	0
$73$	15.9668	1.86877	0.934385	−	0.356264i	$-0.115950\pi$
$73$	15.9668	1.86877	0.934385	+	0.356264i	$0.115950\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9.28811	−1.05848
$78$	0	0
$79$	−8.52746	−0.959414	−0.479707	−	0.877429i	$-0.659257\pi$
$79$	−8.52746	−0.959414	−0.479707	+	0.877429i	$0.659257\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.3381	1.24452	0.622260	−	0.782810i	$-0.286215\pi$
$83$	11.3381	1.24452	0.622260	+	0.782810i	$0.286215\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.122334	0.0131155
$88$	0	0
$89$	14.1326	1.49805	0.749024	−	0.662543i	$-0.230523\pi$
$89$	14.1326	1.49805	0.749024	+	0.662543i	$0.230523\pi$
$90$	0	0
$91$	−5.34980	−0.560812
$92$	0	0
$93$	2.94362	0.305239
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.4828	−1.16590	−0.582949	−	0.812509i	$-0.698101\pi$
$97$	−11.4828	−1.16590	−0.582949	+	0.812509i	$0.698101\pi$
$98$	0	0
$99$	−6.17438	−0.620548
$100$	0	0
$101$	−14.8359	−1.47622	−0.738111	−	0.674679i	$-0.764282\pi$
$101$	−14.8359	−1.47622	−0.738111	+	0.674679i	$0.764282\pi$
$102$	0	0
$103$	19.2645	1.89819	0.949096	−	0.314987i	$-0.102000\pi$
$103$	19.2645	1.89819	0.949096	+	0.314987i	$0.102000\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.63523	0.834799	0.417400	−	0.908723i	$-0.362942\pi$
$107$	8.63523	0.834799	0.417400	+	0.908723i	$0.362942\pi$
$108$	0	0
$109$	−18.6893	−1.79011	−0.895055	−	0.445955i	$-0.852864\pi$
$109$	−18.6893	−1.79011	−0.895055	+	0.445955i	$0.852864\pi$
$110$	0	0
$111$	4.36700	0.414497
$112$	0	0
$113$	6.05638	0.569736	0.284868	−	0.958567i	$-0.408050\pi$
$113$	6.05638	0.569736	0.284868	+	0.958567i	$0.408050\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.55634	−0.328784
$118$	0	0
$119$	0.745682	0.0683566
$120$	0	0
$121$	27.1229	2.46572
$122$	0	0
$123$	−4.25327	−0.383504
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−9.24565	−0.820418	−0.410209	−	0.911991i	$-0.634544\pi$
$127$	−9.24565	−0.820418	−0.410209	+	0.911991i	$0.634544\pi$
$128$	0	0
$129$	3.62663	0.319307
$130$	0	0
$131$	19.0155	1.66140	0.830698	−	0.556724i	$-0.187942\pi$
$131$	19.0155	1.66140	0.830698	+	0.556724i	$0.187942\pi$
$132$	0	0
$133$	0.468851	0.0406546
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.86205	0.671700	0.335850	−	0.941915i	$-0.390976\pi$
$137$	7.86205	0.671700	0.335850	+	0.941915i	$0.390976\pi$
$138$	0	0
$139$	−13.8004	−1.17053	−0.585266	−	0.810842i	$-0.699010\pi$
$139$	−13.8004	−1.17053	−0.585266	+	0.810842i	$0.699010\pi$
$140$	0	0
$141$	−5.28215	−0.444837
$142$	0	0
$143$	21.9582	1.83624
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.73708	0.390708
$148$	0	0
$149$	5.12168	0.419585	0.209792	−	0.977746i	$-0.432721\pi$
$149$	5.12168	0.419585	0.209792	+	0.977746i	$0.432721\pi$
$150$	0	0
$151$	−12.9476	−1.05366	−0.526829	−	0.849972i	$-0.676619\pi$
$151$	−12.9476	−1.05366	−0.526829	+	0.849972i	$0.676619\pi$
$152$	0	0
$153$	0.495700	0.0400750
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.02750	−0.321429	−0.160715	−	0.987001i	$-0.551380\pi$
$157$	−4.02750	−0.321429	−0.160715	+	0.987001i	$0.551380\pi$
$158$	0	0
$159$	8.59978	0.682007
$160$	0	0
$161$	−4.60412	−0.362856
$162$	0	0
$163$	15.6190	1.22338	0.611688	−	0.791099i	$-0.290491\pi$
$163$	15.6190	1.22338	0.611688	+	0.791099i	$0.290491\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.8445	0.916551	0.458276	−	0.888810i	$-0.348467\pi$
$167$	11.8445	0.916551	0.458276	+	0.888810i	$0.348467\pi$
$168$	0	0
$169$	−0.352437	−0.0271105
$170$	0	0
$171$	0.311674	0.0238343
$172$	0	0
$173$	14.8305	1.12754	0.563772	−	0.825930i	$-0.309349\pi$
$173$	14.8305	1.12754	0.563772	+	0.825930i	$0.309349\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.8826	−0.968314
$178$	0	0
$179$	−0.957535	−0.0715695	−0.0357848	−	0.999360i	$-0.511393\pi$
$179$	−0.957535	−0.0715695	−0.0357848	+	0.999360i	$0.511393\pi$
$180$	0	0
$181$	4.14123	0.307815	0.153908	−	0.988085i	$-0.450814\pi$
$181$	4.14123	0.307815	0.153908	+	0.988085i	$0.450814\pi$
$182$	0	0
$183$	11.3209	0.836867
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.06064	−0.223816
$188$	0	0
$189$	−1.50430	−0.109422
$190$	0	0
$191$	−17.6305	−1.27570	−0.637849	−	0.770162i	$-0.720176\pi$
$191$	−17.6305	−1.27570	−0.637849	+	0.770162i	$0.720176\pi$
$192$	0	0
$193$	5.25392	0.378185	0.189093	−	0.981959i	$-0.439445\pi$
$193$	5.25392	0.378185	0.189093	+	0.981959i	$0.439445\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.11597	−0.150756	−0.0753782	−	0.997155i	$-0.524016\pi$
$197$	−2.11597	−0.150756	−0.0753782	+	0.997155i	$0.524016\pi$
$198$	0	0
$199$	9.07029	0.642976	0.321488	−	0.946914i	$-0.395817\pi$
$199$	9.07029	0.642976	0.321488	+	0.946914i	$0.395817\pi$
$200$	0	0
$201$	−13.2414	−0.933975
$202$	0	0
$203$	−0.184026	−0.0129161
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.06064	−0.212729
$208$	0	0
$209$	−1.92439	−0.133113
$210$	0	0
$211$	−19.8149	−1.36412	−0.682058	−	0.731298i	$-0.738915\pi$
$211$	−19.8149	−1.36412	−0.682058	+	0.731298i	$0.738915\pi$
$212$	0	0
$213$	−3.26823	−0.223936
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.42809	−0.300598
$218$	0	0
$219$	−15.9668	−1.07894
$220$	0	0
$221$	−1.76288	−0.118584
$222$	0	0
$223$	6.58214	0.440773	0.220386	−	0.975413i	$-0.429268\pi$
$223$	6.58214	0.440773	0.220386	+	0.975413i	$0.429268\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.7821	0.782006	0.391003	−	0.920389i	$-0.372128\pi$
$227$	11.7821	0.782006	0.391003	+	0.920389i	$0.372128\pi$
$228$	0	0
$229$	11.8899	0.785705	0.392853	−	0.919601i	$-0.371488\pi$
$229$	11.8899	0.785705	0.392853	+	0.919601i	$0.371488\pi$
$230$	0	0
$231$	9.28811	0.611113
$232$	0	0
$233$	−5.05001	−0.330837	−0.165419	−	0.986223i	$-0.552898\pi$
$233$	−5.05001	−0.330837	−0.165419	+	0.986223i	$0.552898\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.52746	0.553918
$238$	0	0
$239$	2.67932	0.173311	0.0866556	−	0.996238i	$-0.472382\pi$
$239$	2.67932	0.173311	0.0866556	+	0.996238i	$0.472382\pi$
$240$	0	0
$241$	12.6561	0.815250	0.407625	−	0.913149i	$-0.366357\pi$
$241$	12.6561	0.815250	0.407625	+	0.913149i	$0.366357\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.10842	−0.0705271
$248$	0	0
$249$	−11.3381	−0.718524
$250$	0	0
$251$	−22.9068	−1.44586	−0.722932	−	0.690919i	$-0.757206\pi$
$251$	−22.9068	−1.44586	−0.722932	+	0.690919i	$0.757206\pi$
$252$	0	0
$253$	18.8975	1.18808
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−27.7764	−1.73264	−0.866322	−	0.499486i	$-0.833522\pi$
$257$	−27.7764	−1.73264	−0.866322	+	0.499486i	$0.833522\pi$
$258$	0	0
$259$	−6.56928	−0.408195
$260$	0	0
$261$	−0.122334	−0.00757225
$262$	0	0
$263$	11.7115	0.722161	0.361081	−	0.932535i	$-0.382408\pi$
$263$	11.7115	0.722161	0.361081	+	0.932535i	$0.382408\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−14.1326	−0.864899
$268$	0	0
$269$	4.20621	0.256457	0.128229	−	0.991745i	$-0.459071\pi$
$269$	4.20621	0.256457	0.128229	+	0.991745i	$0.459071\pi$
$270$	0	0
$271$	−12.5659	−0.763325	−0.381663	−	0.924302i	$-0.624648\pi$
$271$	−12.5659	−0.763325	−0.381663	+	0.924302i	$0.624648\pi$
$272$	0	0
$273$	5.34980	0.323785
$274$	0	0
$275$	0	0
$276$	0	0
$277$	31.8653	1.91460	0.957299	−	0.289100i	$-0.0933559\pi$
$277$	31.8653	1.91460	0.957299	+	0.289100i	$0.0933559\pi$
$278$	0	0
$279$	−2.94362	−0.176230
$280$	0	0
$281$	16.2998	0.972364	0.486182	−	0.873858i	$-0.338389\pi$
$281$	16.2998	0.972364	0.486182	+	0.873858i	$0.338389\pi$
$282$	0	0
$283$	−13.6305	−0.810249	−0.405124	−	0.914262i	$-0.632772\pi$
$283$	−13.6305	−0.810249	−0.405124	+	0.914262i	$0.632772\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.39819	0.377673
$288$	0	0
$289$	−16.7543	−0.985546
$290$	0	0
$291$	11.4828	0.673132
$292$	0	0
$293$	25.2652	1.47601	0.738004	−	0.674796i	$-0.235768\pi$
$293$	25.2652	1.47601	0.738004	+	0.674796i	$0.235768\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6.17438	0.358274
$298$	0	0
$299$	10.8847	0.629477
$300$	0	0
$301$	−5.45554	−0.314452
$302$	0	0
$303$	14.8359	0.852297
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7.01023	−0.400095	−0.200047	−	0.979786i	$-0.564110\pi$
$307$	−7.01023	−0.400095	−0.200047	+	0.979786i	$0.564110\pi$
$308$	0	0
$309$	−19.2645	−1.09592
$310$	0	0
$311$	14.5118	0.822891	0.411446	−	0.911434i	$-0.365024\pi$
$311$	14.5118	0.822891	0.411446	+	0.911434i	$0.365024\pi$
$312$	0	0
$313$	21.9197	1.23897	0.619486	−	0.785008i	$-0.287341\pi$
$313$	21.9197	1.23897	0.619486	+	0.785008i	$0.287341\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−33.7178	−1.89378	−0.946890	−	0.321559i	$-0.895793\pi$
$317$	−33.7178	−1.89378	−0.946890	+	0.321559i	$0.895793\pi$
$318$	0	0
$319$	0.755333	0.0422905
$320$	0	0
$321$	−8.63523	−0.481972
$322$	0	0
$323$	0.154497	0.00859645
$324$	0	0
$325$	0	0
$326$	0	0
$327$	18.6893	1.03352
$328$	0	0
$329$	7.94593	0.438073
$330$	0	0
$331$	33.5950	1.84655	0.923275	−	0.384139i	$-0.125502\pi$
$331$	33.5950	1.84655	0.923275	+	0.384139i	$0.125502\pi$
$332$	0	0
$333$	−4.36700	−0.239310
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.08526	0.168065	0.0840323	−	0.996463i	$-0.473220\pi$
$337$	3.08526	0.168065	0.0840323	+	0.996463i	$0.473220\pi$
$338$	0	0
$339$	−6.05638	−0.328937
$340$	0	0
$341$	18.1750	0.984233
$342$	0	0
$343$	−17.6561	−0.953339
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.5739	0.675005	0.337502	−	0.941325i	$-0.390418\pi$
$347$	12.5739	0.675005	0.337502	+	0.941325i	$0.390418\pi$
$348$	0	0
$349$	13.3100	0.712466	0.356233	−	0.934397i	$-0.384061\pi$
$349$	13.3100	0.712466	0.356233	+	0.934397i	$0.384061\pi$
$350$	0	0
$351$	3.55634	0.189823
$352$	0	0
$353$	−11.2914	−0.600980	−0.300490	−	0.953785i	$-0.597150\pi$
$353$	−11.2914	−0.600980	−0.300490	+	0.953785i	$0.597150\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.745682	−0.0394657
$358$	0	0
$359$	19.4072	1.02427	0.512135	−	0.858905i	$-0.328855\pi$
$359$	19.4072	1.02427	0.512135	+	0.858905i	$0.328855\pi$
$360$	0	0
$361$	−18.9029	−0.994887
$362$	0	0
$363$	−27.1229	−1.42358
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.5703	0.917160	0.458580	−	0.888653i	$-0.348358\pi$
$367$	17.5703	0.917160	0.458580	+	0.888653i	$0.348358\pi$
$368$	0	0
$369$	4.25327	0.221416
$370$	0	0
$371$	−12.9367	−0.671637
$372$	0	0
$373$	−28.2065	−1.46048	−0.730238	−	0.683193i	$-0.760591\pi$
$373$	−28.2065	−1.46048	−0.730238	+	0.683193i	$0.760591\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.435060	0.0224067
$378$	0	0
$379$	3.62822	0.186369	0.0931845	−	0.995649i	$-0.470295\pi$
$379$	3.62822	0.186369	0.0931845	+	0.995649i	$0.470295\pi$
$380$	0	0
$381$	9.24565	0.473669
$382$	0	0
$383$	−13.8348	−0.706925	−0.353463	−	0.935449i	$-0.614996\pi$
$383$	−13.8348	−0.706925	−0.353463	+	0.935449i	$0.614996\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.62663	−0.184352
$388$	0	0
$389$	−20.9014	−1.05974	−0.529871	−	0.848079i	$-0.677759\pi$
$389$	−20.9014	−1.05974	−0.529871	+	0.848079i	$0.677759\pi$
$390$	0	0
$391$	−1.51716	−0.0767261
$392$	0	0
$393$	−19.0155	−0.959207
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−15.4522	−0.775523	−0.387761	−	0.921760i	$-0.626752\pi$
$397$	−15.4522	−0.775523	−0.387761	+	0.921760i	$0.626752\pi$
$398$	0	0
$399$	−0.468851	−0.0234719
$400$	0	0
$401$	2.25590	0.112654	0.0563271	−	0.998412i	$-0.482061\pi$
$401$	2.25590	0.112654	0.0563271	+	0.998412i	$0.482061\pi$
$402$	0	0
$403$	10.4685	0.521474
$404$	0	0
$405$	0	0
$406$	0	0
$407$	26.9635	1.33653
$408$	0	0
$409$	−2.33260	−0.115340	−0.0576699	−	0.998336i	$-0.518367\pi$
$409$	−2.33260	−0.115340	−0.0576699	+	0.998336i	$0.518367\pi$
$410$	0	0
$411$	−7.86205	−0.387806
$412$	0	0
$413$	19.3793	0.953591
$414$	0	0
$415$	0	0
$416$	0	0
$417$	13.8004	0.675806
$418$	0	0
$419$	−5.04076	−0.246257	−0.123129	−	0.992391i	$-0.539293\pi$
$419$	−5.04076	−0.246257	−0.123129	+	0.992391i	$0.539293\pi$
$420$	0	0
$421$	12.9386	0.630587	0.315293	−	0.948994i	$-0.397897\pi$
$421$	12.9386	0.630587	0.315293	+	0.948994i	$0.397897\pi$
$422$	0	0
$423$	5.28215	0.256827
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−17.0301	−0.824142
$428$	0	0
$429$	−21.9582	−1.06015
$430$	0	0
$431$	1.56501	0.0753841	0.0376920	−	0.999289i	$-0.487999\pi$
$431$	1.56501	0.0753841	0.0376920	+	0.999289i	$0.487999\pi$
$432$	0	0
$433$	25.8729	1.24337	0.621687	−	0.783266i	$-0.286448\pi$
$433$	25.8729	1.24337	0.621687	+	0.783266i	$0.286448\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.953923	−0.0456323
$438$	0	0
$439$	16.3713	0.781358	0.390679	−	0.920527i	$-0.372240\pi$
$439$	16.3713	0.781358	0.390679	+	0.920527i	$0.372240\pi$
$440$	0	0
$441$	−4.73708	−0.225575
$442$	0	0
$443$	28.1180	1.33593	0.667963	−	0.744194i	$-0.267166\pi$
$443$	28.1180	1.33593	0.667963	+	0.744194i	$0.267166\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5.12168	−0.242247
$448$	0	0
$449$	−2.65681	−0.125383	−0.0626914	−	0.998033i	$-0.519968\pi$
$449$	−2.65681	−0.125383	−0.0626914	+	0.998033i	$0.519968\pi$
$450$	0	0
$451$	−26.2613	−1.23659
$452$	0	0
$453$	12.9476	0.608329
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.321574	−0.0150426	−0.00752129	−	0.999972i	$-0.502394\pi$
$457$	−0.321574	−0.0150426	−0.00752129	+	0.999972i	$0.502394\pi$
$458$	0	0
$459$	−0.495700	−0.0231373
$460$	0	0
$461$	−7.68562	−0.357955	−0.178977	−	0.983853i	$-0.557279\pi$
$461$	−7.68562	−0.357955	−0.178977	+	0.983853i	$0.557279\pi$
$462$	0	0
$463$	19.1041	0.887842	0.443921	−	0.896066i	$-0.353587\pi$
$463$	19.1041	0.887842	0.443921	+	0.896066i	$0.353587\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.41944	−0.297056	−0.148528	−	0.988908i	$-0.547454\pi$
$467$	−6.41944	−0.297056	−0.148528	+	0.988908i	$0.547454\pi$
$468$	0	0
$469$	19.9190	0.919774
$470$	0	0
$471$	4.02750	0.185577
$472$	0	0
$473$	22.3922	1.02959
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−8.59978	−0.393757
$478$	0	0
$479$	33.5609	1.53344	0.766719	−	0.641983i	$-0.221888\pi$
$479$	33.5609	1.53344	0.766719	+	0.641983i	$0.221888\pi$
$480$	0	0
$481$	15.5305	0.708132
$482$	0	0
$483$	4.60412	0.209495
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−11.9105	−0.539715	−0.269858	−	0.962900i	$-0.586977\pi$
$487$	−11.9105	−0.539715	−0.269858	+	0.962900i	$0.586977\pi$
$488$	0	0
$489$	−15.6190	−0.706316
$490$	0	0
$491$	15.6130	0.704607	0.352303	−	0.935886i	$-0.385398\pi$
$491$	15.6130	0.704607	0.352303	+	0.935886i	$0.385398\pi$
$492$	0	0
$493$	−0.0606408	−0.00273112
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.91640	0.220531
$498$	0	0
$499$	30.9130	1.38386	0.691929	−	0.721966i	$-0.256761\pi$
$499$	30.9130	1.38386	0.691929	+	0.721966i	$0.256761\pi$
$500$	0	0
$501$	−11.8445	−0.529171
$502$	0	0
$503$	31.0396	1.38399	0.691994	−	0.721903i	$-0.256732\pi$
$503$	31.0396	1.38399	0.691994	+	0.721903i	$0.256732\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.352437	0.0156523
$508$	0	0
$509$	−13.3428	−0.591409	−0.295704	−	0.955280i	$-0.595554\pi$
$509$	−13.3428	−0.591409	−0.295704	+	0.955280i	$0.595554\pi$
$510$	0	0
$511$	24.0188	1.06253
$512$	0	0
$513$	−0.311674	−0.0137608
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−32.6139	−1.43436
$518$	0	0
$519$	−14.8305	−0.650988
$520$	0	0
$521$	26.5176	1.16176	0.580879	−	0.813990i	$-0.302709\pi$
$521$	26.5176	1.16176	0.580879	+	0.813990i	$0.302709\pi$
$522$	0	0
$523$	−17.4147	−0.761492	−0.380746	−	0.924680i	$-0.624333\pi$
$523$	−17.4147	−0.761492	−0.380746	+	0.924680i	$0.624333\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.45915	−0.0635618
$528$	0	0
$529$	−13.6325	−0.592716
$530$	0	0
$531$	12.8826	0.559056
$532$	0	0
$533$	−15.1261	−0.655183
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.957535	0.0413207
$538$	0	0
$539$	29.2485	1.25982
$540$	0	0
$541$	5.46682	0.235037	0.117519	−	0.993071i	$-0.462506\pi$
$541$	5.46682	0.235037	0.117519	+	0.993071i	$0.462506\pi$
$542$	0	0
$543$	−4.14123	−0.177717
$544$	0	0
$545$	0	0
$546$	0	0
$547$	21.6459	0.925510	0.462755	−	0.886486i	$-0.346861\pi$
$547$	21.6459	0.925510	0.462755	+	0.886486i	$0.346861\pi$
$548$	0	0
$549$	−11.3209	−0.483165
$550$	0	0
$551$	−0.0381282	−0.00162432
$552$	0	0
$553$	−12.8279	−0.545496
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.58830	−0.109670	−0.0548349	−	0.998495i	$-0.517463\pi$
$557$	−2.58830	−0.109670	−0.0548349	+	0.998495i	$0.517463\pi$
$558$	0	0
$559$	12.8975	0.545508
$560$	0	0
$561$	3.06064	0.129220
$562$	0	0
$563$	19.8939	0.838426	0.419213	−	0.907888i	$-0.362306\pi$
$563$	19.8939	0.838426	0.419213	+	0.907888i	$0.362306\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.50430	0.0631746
$568$	0	0
$569$	32.4303	1.35955	0.679775	−	0.733421i	$-0.262077\pi$
$569$	32.4303	1.35955	0.679775	+	0.733421i	$0.262077\pi$
$570$	0	0
$571$	13.7182	0.574088	0.287044	−	0.957917i	$-0.407327\pi$
$571$	13.7182	0.574088	0.287044	+	0.957917i	$0.407327\pi$
$572$	0	0
$573$	17.6305	0.736524
$574$	0	0
$575$	0	0
$576$	0	0
$577$	24.6432	1.02591	0.512955	−	0.858415i	$-0.328551\pi$
$577$	24.6432	1.02591	0.512955	+	0.858415i	$0.328551\pi$
$578$	0	0
$579$	−5.25392	−0.218345
$580$	0	0
$581$	17.0559	0.707599
$582$	0	0
$583$	53.0983	2.19911
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.32165	0.343471	0.171736	−	0.985143i	$-0.445062\pi$
$587$	8.32165	0.343471	0.171736	+	0.985143i	$0.445062\pi$
$588$	0	0
$589$	−0.917451	−0.0378029
$590$	0	0
$591$	2.11597	0.0870393
$592$	0	0
$593$	21.7904	0.894826	0.447413	−	0.894328i	$-0.352346\pi$
$593$	21.7904	0.894826	0.447413	+	0.894328i	$0.352346\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−9.07029	−0.371222
$598$	0	0
$599$	10.6482	0.435074	0.217537	−	0.976052i	$-0.430198\pi$
$599$	10.6482	0.435074	0.217537	+	0.976052i	$0.430198\pi$
$600$	0	0
$601$	2.60740	0.106358	0.0531791	−	0.998585i	$-0.483065\pi$
$601$	2.60740	0.106358	0.0531791	+	0.998585i	$0.483065\pi$
$602$	0	0
$603$	13.2414	0.539231
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.9820	1.33870	0.669349	−	0.742948i	$-0.266573\pi$
$607$	32.9820	1.33870	0.669349	+	0.742948i	$0.266573\pi$
$608$	0	0
$609$	0.184026	0.00745712
$610$	0	0
$611$	−18.7851	−0.759964
$612$	0	0
$613$	−18.7891	−0.758886	−0.379443	−	0.925215i	$-0.623884\pi$
$613$	−18.7891	−0.758886	−0.379443	+	0.925215i	$0.623884\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	39.3278	1.58328	0.791639	−	0.610990i	$-0.209228\pi$
$617$	39.3278	1.58328	0.791639	+	0.610990i	$0.209228\pi$
$618$	0	0
$619$	−16.0365	−0.644560	−0.322280	−	0.946644i	$-0.604449\pi$
$619$	−16.0365	−0.644560	−0.322280	+	0.946644i	$0.604449\pi$
$620$	0	0
$621$	3.06064	0.122819
$622$	0	0
$623$	21.2596	0.851748
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.92439	0.0768529
$628$	0	0
$629$	−2.16472	−0.0863132
$630$	0	0
$631$	4.82685	0.192154	0.0960770	−	0.995374i	$-0.469371\pi$
$631$	4.82685	0.192154	0.0960770	+	0.995374i	$0.469371\pi$
$632$	0	0
$633$	19.8149	0.787572
$634$	0	0
$635$	0	0
$636$	0	0
$637$	16.8467	0.667490
$638$	0	0
$639$	3.26823	0.129289
$640$	0	0
$641$	−36.5645	−1.44421	−0.722106	−	0.691782i	$-0.756826\pi$
$641$	−36.5645	−1.44421	−0.722106	+	0.691782i	$0.756826\pi$
$642$	0	0
$643$	−18.3769	−0.724714	−0.362357	−	0.932039i	$-0.618028\pi$
$643$	−18.3769	−0.724714	−0.362357	+	0.932039i	$0.618028\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−41.1338	−1.61714	−0.808568	−	0.588403i	$-0.799757\pi$
$647$	−41.1338	−1.61714	−0.808568	+	0.588403i	$0.799757\pi$
$648$	0	0
$649$	−79.5419	−3.12229
$650$	0	0
$651$	4.42809	0.173550
$652$	0	0
$653$	−21.9447	−0.858761	−0.429381	−	0.903124i	$-0.641268\pi$
$653$	−21.9447	−0.858761	−0.429381	+	0.903124i	$0.641268\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	15.9668	0.622924
$658$	0	0
$659$	21.2048	0.826020	0.413010	−	0.910726i	$-0.364477\pi$
$659$	21.2048	0.826020	0.413010	+	0.910726i	$0.364477\pi$
$660$	0	0
$661$	17.9442	0.697948	0.348974	−	0.937132i	$-0.386530\pi$
$661$	17.9442	0.697948	0.348974	+	0.937132i	$0.386530\pi$
$662$	0	0
$663$	1.76288	0.0684646
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.374419	0.0144976
$668$	0	0
$669$	−6.58214	−0.254480
$670$	0	0
$671$	69.8996	2.69845
$672$	0	0
$673$	34.3014	1.32222	0.661111	−	0.750288i	$-0.270085\pi$
$673$	34.3014	1.32222	0.661111	+	0.750288i	$0.270085\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.14695	−0.159380	−0.0796901	−	0.996820i	$-0.525393\pi$
$677$	−4.14695	−0.159380	−0.0796901	+	0.996820i	$0.525393\pi$
$678$	0	0
$679$	−17.2735	−0.662897
$680$	0	0
$681$	−11.7821	−0.451491
$682$	0	0
$683$	2.11840	0.0810583	0.0405291	−	0.999178i	$-0.487096\pi$
$683$	2.11840	0.0810583	0.0405291	+	0.999178i	$0.487096\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−11.8899	−0.453627
$688$	0	0
$689$	30.5838	1.16515
$690$	0	0
$691$	−3.97453	−0.151198	−0.0755991	−	0.997138i	$-0.524087\pi$
$691$	−3.97453	−0.151198	−0.0755991	+	0.997138i	$0.524087\pi$
$692$	0	0
$693$	−9.28811	−0.352826
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.10835	0.0798593
$698$	0	0
$699$	5.05001	0.191009
$700$	0	0
$701$	−10.1954	−0.385076	−0.192538	−	0.981289i	$-0.561672\pi$
$701$	−10.1954	−0.385076	−0.192538	+	0.981289i	$0.561672\pi$
$702$	0	0
$703$	−1.36108	−0.0513342
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−22.3176	−0.839338
$708$	0	0
$709$	29.8413	1.12071	0.560357	−	0.828251i	$-0.310664\pi$
$709$	29.8413	1.12071	0.560357	+	0.828251i	$0.310664\pi$
$710$	0	0
$711$	−8.52746	−0.319805
$712$	0	0
$713$	9.00937	0.337404
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−2.67932	−0.100061
$718$	0	0
$719$	7.43434	0.277254	0.138627	−	0.990345i	$-0.455731\pi$
$719$	7.43434	0.277254	0.138627	+	0.990345i	$0.455731\pi$
$720$	0	0
$721$	28.9796	1.07926
$722$	0	0
$723$	−12.6561	−0.470685
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−15.9195	−0.590420	−0.295210	−	0.955432i	$-0.595390\pi$
$727$	−15.9195	−0.590420	−0.295210	+	0.955432i	$0.595390\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.79772	−0.0664912
$732$	0	0
$733$	30.6103	1.13062	0.565308	−	0.824880i	$-0.308757\pi$
$733$	30.6103	1.13062	0.565308	+	0.824880i	$0.308757\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−81.7573	−3.01157
$738$	0	0
$739$	25.9379	0.954141	0.477071	−	0.878865i	$-0.341699\pi$
$739$	25.9379	0.954141	0.477071	+	0.878865i	$0.341699\pi$
$740$	0	0
$741$	1.10842	0.0407188
$742$	0	0
$743$	17.2136	0.631504	0.315752	−	0.948842i	$-0.397743\pi$
$743$	17.2136	0.631504	0.315752	+	0.948842i	$0.397743\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	11.3381	0.414840
$748$	0	0
$749$	12.9900	0.474643
$750$	0	0
$751$	10.4020	0.379576	0.189788	−	0.981825i	$-0.439220\pi$
$751$	10.4020	0.379576	0.189788	+	0.981825i	$0.439220\pi$
$752$	0	0
$753$	22.9068	0.834770
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−16.9118	−0.614669	−0.307335	−	0.951601i	$-0.599437\pi$
$757$	−16.9118	−0.614669	−0.307335	+	0.951601i	$0.599437\pi$
$758$	0	0
$759$	−18.8975	−0.685937
$760$	0	0
$761$	0.522347	0.0189351	0.00946753	−	0.999955i	$-0.496986\pi$
$761$	0.522347	0.0189351	0.00946753	+	0.999955i	$0.496986\pi$
$762$	0	0
$763$	−28.1143	−1.01781
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−45.8148	−1.65428
$768$	0	0
$769$	−38.0791	−1.37317	−0.686584	−	0.727050i	$-0.740891\pi$
$769$	−38.0791	−1.37317	−0.686584	+	0.727050i	$0.740891\pi$
$770$	0	0
$771$	27.7764	1.00034
$772$	0	0
$773$	−22.6913	−0.816151	−0.408075	−	0.912948i	$-0.633800\pi$
$773$	−22.6913	−0.816151	−0.408075	+	0.912948i	$0.633800\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6.56928	0.235671
$778$	0	0
$779$	1.32563	0.0474957
$780$	0	0
$781$	−20.1793	−0.722072
$782$	0	0
$783$	0.122334	0.00437184
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2.27911	0.0812415	0.0406207	−	0.999175i	$-0.487066\pi$
$787$	2.27911	0.0812415	0.0406207	+	0.999175i	$0.487066\pi$
$788$	0	0
$789$	−11.7115	−0.416940
$790$	0	0
$791$	9.11061	0.323936
$792$	0	0
$793$	40.2611	1.42971
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19.6409	0.695715	0.347858	−	0.937547i	$-0.386909\pi$
$797$	19.6409	0.695715	0.347858	+	0.937547i	$0.386909\pi$
$798$	0	0
$799$	2.61836	0.0926310
$800$	0	0
$801$	14.1326	0.499349
$802$	0	0
$803$	−98.5849	−3.47899
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.20621	−0.148066
$808$	0	0
$809$	11.1951	0.393599	0.196799	−	0.980444i	$-0.436945\pi$
$809$	11.1951	0.393599	0.196799	+	0.980444i	$0.436945\pi$
$810$	0	0
$811$	−19.2989	−0.677677	−0.338839	−	0.940844i	$-0.610034\pi$
$811$	−19.2989	−0.677677	−0.338839	+	0.940844i	$0.610034\pi$
$812$	0	0
$813$	12.5659	0.440706
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1.13033	−0.0395452
$818$	0	0
$819$	−5.34980	−0.186937
$820$	0	0
$821$	−48.1255	−1.67959	−0.839797	−	0.542901i	$-0.817326\pi$
$821$	−48.1255	−1.67959	−0.839797	+	0.542901i	$0.817326\pi$
$822$	0	0
$823$	34.9012	1.21658	0.608289	−	0.793716i	$-0.291856\pi$
$823$	34.9012	1.21658	0.608289	+	0.793716i	$0.291856\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.1640	0.805491	0.402746	−	0.915312i	$-0.368056\pi$
$827$	23.1640	0.805491	0.402746	+	0.915312i	$0.368056\pi$
$828$	0	0
$829$	21.7129	0.754121	0.377060	−	0.926189i	$-0.376935\pi$
$829$	21.7129	0.754121	0.377060	+	0.926189i	$0.376935\pi$
$830$	0	0
$831$	−31.8653	−1.10539
$832$	0	0
$833$	−2.34817	−0.0813594
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.94362	0.101746
$838$	0	0
$839$	10.3890	0.358668	0.179334	−	0.983788i	$-0.442606\pi$
$839$	10.3890	0.358668	0.179334	+	0.983788i	$0.442606\pi$
$840$	0	0
$841$	−28.9850	−0.999484
$842$	0	0
$843$	−16.2998	−0.561395
$844$	0	0
$845$	0	0
$846$	0	0
$847$	40.8010	1.40194
$848$	0	0
$849$	13.6305	0.467797
$850$	0	0
$851$	13.3658	0.458174
$852$	0	0
$853$	33.6101	1.15079	0.575395	−	0.817876i	$-0.304848\pi$
$853$	33.6101	1.15079	0.575395	+	0.817876i	$0.304848\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.0078	1.57160	0.785799	−	0.618482i	$-0.212252\pi$
$857$	46.0078	1.57160	0.785799	+	0.618482i	$0.212252\pi$
$858$	0	0
$859$	−30.3516	−1.03558	−0.517791	−	0.855507i	$-0.673245\pi$
$859$	−30.3516	−1.03558	−0.517791	+	0.855507i	$0.673245\pi$
$860$	0	0
$861$	−6.39819	−0.218050
$862$	0	0
$863$	−7.32052	−0.249193	−0.124597	−	0.992207i	$-0.539764\pi$
$863$	−7.32052	−0.249193	−0.124597	+	0.992207i	$0.539764\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.7543	0.569005
$868$	0	0
$869$	52.6518	1.78609
$870$	0	0
$871$	−47.0909	−1.59561
$872$	0	0
$873$	−11.4828	−0.388633
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−46.5437	−1.57167	−0.785834	−	0.618438i	$-0.787766\pi$
$877$	−46.5437	−1.57167	−0.785834	+	0.618438i	$0.787766\pi$
$878$	0	0
$879$	−25.2652	−0.852174
$880$	0	0
$881$	−14.9721	−0.504423	−0.252211	−	0.967672i	$-0.581158\pi$
$881$	−14.9721	−0.504423	−0.252211	+	0.967672i	$0.581158\pi$
$882$	0	0
$883$	−8.74240	−0.294205	−0.147103	−	0.989121i	$-0.546995\pi$
$883$	−8.74240	−0.294205	−0.147103	+	0.989121i	$0.546995\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.4865	0.553563	0.276782	−	0.960933i	$-0.410732\pi$
$887$	16.4865	0.553563	0.276782	+	0.960933i	$0.410732\pi$
$888$	0	0
$889$	−13.9082	−0.466467
$890$	0	0
$891$	−6.17438	−0.206849
$892$	0	0
$893$	1.64631	0.0550916
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−10.8847	−0.363429
$898$	0	0
$899$	0.360104	0.0120101
$900$	0	0
$901$	−4.26292	−0.142018
$902$	0	0
$903$	5.45554	0.181549
$904$	0	0
$905$	0	0
$906$	0	0
$907$	26.0243	0.864124	0.432062	−	0.901844i	$-0.357786\pi$
$907$	26.0243	0.864124	0.432062	+	0.901844i	$0.357786\pi$
$908$	0	0
$909$	−14.8359	−0.492074
$910$	0	0
$911$	−51.6447	−1.71107	−0.855533	−	0.517748i	$-0.826771\pi$
$911$	−51.6447	−1.71107	−0.855533	+	0.517748i	$0.826771\pi$
$912$	0	0
$913$	−70.0058	−2.31685
$914$	0	0
$915$	0	0
$916$	0	0
$917$	28.6051	0.944623
$918$	0	0
$919$	−27.6307	−0.911452	−0.455726	−	0.890120i	$-0.650620\pi$
$919$	−27.6307	−0.911452	−0.455726	+	0.890120i	$0.650620\pi$
$920$	0	0
$921$	7.01023	0.230995
$922$	0	0
$923$	−11.6229	−0.382574
$924$	0	0
$925$	0	0
$926$	0	0
$927$	19.2645	0.632731
$928$	0	0
$929$	53.7012	1.76188	0.880939	−	0.473230i	$-0.156912\pi$
$929$	53.7012	1.76188	0.880939	+	0.473230i	$0.156912\pi$
$930$	0	0
$931$	−1.47643	−0.0483879
$932$	0	0
$933$	−14.5118	−0.475097
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−30.3213	−0.990552	−0.495276	−	0.868736i	$-0.664933\pi$
$937$	−30.3213	−0.990552	−0.495276	+	0.868736i	$0.664933\pi$
$938$	0	0
$939$	−21.9197	−0.715321
$940$	0	0
$941$	47.3682	1.54416	0.772080	−	0.635525i	$-0.219216\pi$
$941$	47.3682	1.54416	0.772080	+	0.635525i	$0.219216\pi$
$942$	0	0
$943$	−13.0177	−0.423915
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21.6419	−0.703266	−0.351633	−	0.936138i	$-0.614373\pi$
$947$	−21.6419	−0.703266	−0.351633	+	0.936138i	$0.614373\pi$
$948$	0	0
$949$	−56.7833	−1.84327
$950$	0	0
$951$	33.7178	1.09337
$952$	0	0
$953$	−45.8917	−1.48658	−0.743289	−	0.668970i	$-0.766735\pi$
$953$	−45.8917	−1.48658	−0.743289	+	0.668970i	$0.766735\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−0.755333	−0.0244164
$958$	0	0
$959$	11.8269	0.381910
$960$	0	0
$961$	−22.3351	−0.720487
$962$	0	0
$963$	8.63523	0.278266
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−14.9696	−0.481391	−0.240695	−	0.970601i	$-0.577375\pi$
$967$	−14.9696	−0.481391	−0.240695	+	0.970601i	$0.577375\pi$
$968$	0	0
$969$	−0.154497	−0.00496316
$970$	0	0
$971$	−22.3795	−0.718191	−0.359096	−	0.933301i	$-0.616915\pi$
$971$	−22.3795	−0.718191	−0.359096	+	0.933301i	$0.616915\pi$
$972$	0	0
$973$	−20.7599	−0.665531
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−40.3686	−1.29150	−0.645752	−	0.763547i	$-0.723456\pi$
$977$	−40.3686	−1.29150	−0.645752	+	0.763547i	$0.723456\pi$
$978$	0	0
$979$	−87.2597	−2.78883
$980$	0	0
$981$	−18.6893	−0.596704
$982$	0	0
$983$	−23.3730	−0.745482	−0.372741	−	0.927935i	$-0.621582\pi$
$983$	−23.3730	−0.745482	−0.372741	+	0.927935i	$0.621582\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−7.94593	−0.252922
$988$	0	0
$989$	11.0998	0.352954
$990$	0	0
$991$	33.8454	1.07514	0.537568	−	0.843221i	$-0.319343\pi$
$991$	33.8454	1.07514	0.537568	+	0.843221i	$0.319343\pi$
$992$	0	0
$993$	−33.5950	−1.06611
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.54503	0.175613	0.0878065	−	0.996138i	$-0.472014\pi$
$997$	5.54503	0.175613	0.0878065	+	0.996138i	$0.472014\pi$
$998$	0	0
$999$	4.36700	0.138166

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.a.e.1.3		4
5.2	odd	4		7500.2.d.c.1249.7		8
5.3	odd	4		7500.2.d.c.1249.2		8
5.4	even	2		7500.2.a.f.1.2		4
25.3	odd	20		1500.2.o.b.49.3		16
25.4	even	10		1500.2.m.a.1201.1		8
25.6	even	5		300.2.m.b.61.2	✓	8
25.8	odd	20		1500.2.o.b.949.1		16
25.17	odd	20		1500.2.o.b.949.4		16
25.19	even	10		1500.2.m.a.301.1		8
25.21	even	5		300.2.m.b.241.2	yes	8
25.22	odd	20		1500.2.o.b.49.2		16
75.56	odd	10		900.2.n.b.361.1		8
75.71	odd	10		900.2.n.b.541.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.m.b.61.2	✓	8	25.6	even	5
300.2.m.b.241.2	yes	8	25.21	even	5
900.2.n.b.361.1		8	75.56	odd	10
900.2.n.b.541.1		8	75.71	odd	10
1500.2.m.a.301.1		8	25.19	even	10
1500.2.m.a.1201.1		8	25.4	even	10
1500.2.o.b.49.2		16	25.22	odd	20
1500.2.o.b.49.3		16	25.3	odd	20
1500.2.o.b.949.1		16	25.8	odd	20
1500.2.o.b.949.4		16	25.17	odd	20
7500.2.a.e.1.3		4	1.1	even	1	trivial
7500.2.a.f.1.2		4	5.4	even	2
7500.2.d.c.1249.2		8	5.3	odd	4
7500.2.d.c.1249.7		8	5.2	odd	4

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:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.50430 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.50430 q^{7} +1.00000 q^{9} -6.17438 q^{11} -3.55634 q^{13} +0.495700 q^{17} +0.311674 q^{19} -1.50430 q^{21} -3.06064 q^{23} -1.00000 q^{27} -0.122334 q^{29} -2.94362 q^{31} +6.17438 q^{33} -4.36700 q^{37} +3.55634 q^{39} +4.25327 q^{41} -3.62663 q^{43} +5.28215 q^{47} -4.73708 q^{49} -0.495700 q^{51} -8.59978 q^{53} -0.311674 q^{57} +12.8826 q^{59} -11.3209 q^{61} +1.50430 q^{63} +13.2414 q^{67} +3.06064 q^{69} +3.26823 q^{71} +15.9668 q^{73} -9.28811 q^{77} -8.52746 q^{79} +1.00000 q^{81} +11.3381 q^{83} +0.122334 q^{87} +14.1326 q^{89} -5.34980 q^{91} +2.94362 q^{93} -11.4828 q^{97} -6.17438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9} - q^{11} + 5 q^{13} + 4 q^{17} - 5 q^{19} - 4 q^{21} + 9 q^{23} - 4 q^{27} + 6 q^{29} + 11 q^{31} + q^{33} + 2 q^{37} - 5 q^{39} - 6 q^{43} + 16 q^{47} - 4 q^{49} - 4 q^{51} - 2 q^{53} + 5 q^{57} + q^{59} - 22 q^{61} + 4 q^{63} + 36 q^{67} - 9 q^{69} + 20 q^{71} + 12 q^{73} - 11 q^{77} - 3 q^{79} + 4 q^{81} + 14 q^{83} - 6 q^{87} - 15 q^{89} - 10 q^{91} - 11 q^{93} - 12 q^{97} - q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^7 + 4 * q^9 - q^11 + 5 * q^13 + 4 * q^17 - 5 * q^19 - 4 * q^21 + 9 * q^23 - 4 * q^27 + 6 * q^29 + 11 * q^31 + q^33 + 2 * q^37 - 5 * q^39 - 6 * q^43 + 16 * q^47 - 4 * q^49 - 4 * q^51 - 2 * q^53 + 5 * q^57 + q^59 - 22 * q^61 + 4 * q^63 + 36 * q^67 - 9 * q^69 + 20 * q^71 + 12 * q^73 - 11 * q^77 - 3 * q^79 + 4 * q^81 + 14 * q^83 - 6 * q^87 - 15 * q^89 - 10 * q^91 - 11 * q^93 - 12 * q^97 - q^99

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