LMFDB - Embedded newform 7350.2.a.dt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7350,2,Mod(1,7350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7350, 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("7350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7350.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:	$58.6900454856$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.10304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 8 $ x^4 - 2x^3 - 7x^2 + 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1470)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.692297$ of defining polynomial
Character	$\chi$	$=$	7350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.39327 q^{11} -1.00000 q^{12} -3.80748 q^{13} +1.00000 q^{16} -1.97038 q^{17} +1.00000 q^{18} +4.17113 q^{19} -2.39327 q^{22} +8.17113 q^{23} -1.00000 q^{24} -3.80748 q^{26} -1.00000 q^{27} -9.16246 q^{29} -1.60673 q^{31} +1.00000 q^{32} +2.39327 q^{33} -1.97038 q^{34} +1.00000 q^{36} +1.26358 q^{37} +4.17113 q^{38} +3.80748 q^{39} +3.19208 q^{41} +4.90755 q^{43} -2.39327 q^{44} +8.17113 q^{46} -3.00868 q^{47} -1.00000 q^{48} +1.97038 q^{51} -3.80748 q^{52} +12.1711 q^{53} -1.00000 q^{54} -4.17113 q^{57} -9.16246 q^{58} -13.0414 q^{59} -13.0910 q^{61} -1.60673 q^{62} +1.00000 q^{64} +2.39327 q^{66} -5.77786 q^{67} -1.97038 q^{68} -8.17113 q^{69} +6.94032 q^{71} +1.00000 q^{72} +0.506664 q^{73} +1.26358 q^{74} +4.17113 q^{76} +3.80748 q^{78} -12.4260 q^{79} +1.00000 q^{81} +3.19208 q^{82} +1.89887 q^{83} +4.90755 q^{86} +9.16246 q^{87} -2.39327 q^{88} -1.97949 q^{89} +8.17113 q^{92} +1.60673 q^{93} -3.00868 q^{94} -1.00000 q^{96} -12.0624 q^{97} -2.39327 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^6 + 4 * q^8 + 4 * q^9	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} + 4 q^{8} + 4 q^{9} - 4 q^{12} + 4 q^{16} - 4 q^{17} + 4 q^{18} - 12 q^{19} + 4 q^{23} - 4 q^{24} - 4 q^{27} - 8 q^{29} - 16 q^{31} + 4 q^{32} - 4 q^{34} + 4 q^{36} - 8 q^{37} - 12 q^{38} - 12 q^{41} + 4 q^{43} + 4 q^{46} - 12 q^{47} - 4 q^{48} + 4 q^{51} + 20 q^{53} - 4 q^{54} + 12 q^{57} - 8 q^{58} - 20 q^{59} - 12 q^{61} - 16 q^{62} + 4 q^{64} - 4 q^{67} - 4 q^{68} - 4 q^{69} - 20 q^{71} + 4 q^{72} + 12 q^{73} - 8 q^{74} - 12 q^{76} - 8 q^{79} + 4 q^{81} - 12 q^{82} - 8 q^{83} + 4 q^{86} + 8 q^{87} - 44 q^{89} + 4 q^{92} + 16 q^{93} - 12 q^{94} - 4 q^{96} - 20 q^{97}+O(q^{100}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^6 + 4 * q^8 + 4 * q^9 - 4 * q^12 + 4 * q^16 - 4 * q^17 + 4 * q^18 - 12 * q^19 + 4 * q^23 - 4 * q^24 - 4 * q^27 - 8 * q^29 - 16 * q^31 + 4 * q^32 - 4 * q^34 + 4 * q^36 - 8 * q^37 - 12 * q^38 - 12 * q^41 + 4 * q^43 + 4 * q^46 - 12 * q^47 - 4 * q^48 + 4 * q^51 + 20 * q^53 - 4 * q^54 + 12 * q^57 - 8 * q^58 - 20 * q^59 - 12 * q^61 - 16 * q^62 + 4 * q^64 - 4 * q^67 - 4 * q^68 - 4 * q^69 - 20 * q^71 + 4 * q^72 + 12 * q^73 - 8 * q^74 - 12 * q^76 - 8 * q^79 + 4 * q^81 - 12 * q^82 - 8 * q^83 + 4 * q^86 + 8 * q^87 - 44 * q^89 + 4 * q^92 + 16 * q^93 - 12 * q^94 - 4 * q^96 - 20 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.39327	−0.721598	−0.360799	−	0.932644i	$-0.617496\pi$
$11$	−2.39327	−0.721598	−0.360799	+	0.932644i	$0.617496\pi$
$12$	−1.00000	−0.288675
$13$	−3.80748	−1.05601	−0.528003	−	0.849243i	$-0.677059\pi$
$13$	−3.80748	−1.05601	−0.528003	+	0.849243i	$0.677059\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.97038	−0.477888	−0.238944	−	0.971033i	$-0.576801\pi$
$17$	−1.97038	−0.477888	−0.238944	+	0.971033i	$0.576801\pi$
$18$	1.00000	0.235702
$19$	4.17113	0.956924	0.478462	−	0.878108i	$-0.341194\pi$
$19$	4.17113	0.956924	0.478462	+	0.878108i	$0.341194\pi$
$20$	0	0
$21$	0	0
$22$	−2.39327	−0.510247
$23$	8.17113	1.70380	0.851900	−	0.523705i	$-0.175451\pi$
$23$	8.17113	1.70380	0.851900	+	0.523705i	$0.175451\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−3.80748	−0.746709
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−9.16246	−1.70143	−0.850713	−	0.525631i	$-0.823829\pi$
$29$	−9.16246	−1.70143	−0.850713	+	0.525631i	$0.823829\pi$
$30$	0	0
$31$	−1.60673	−0.288577	−0.144289	−	0.989536i	$-0.546089\pi$
$31$	−1.60673	−0.288577	−0.144289	+	0.989536i	$0.546089\pi$
$32$	1.00000	0.176777
$33$	2.39327	0.416615
$34$	−1.97038	−0.337918
$35$	0	0
$36$	1.00000	0.166667
$37$	1.26358	0.207732	0.103866	−	0.994591i	$-0.466879\pi$
$37$	1.26358	0.207732	0.103866	+	0.994591i	$0.466879\pi$
$38$	4.17113	0.676647
$39$	3.80748	0.609685
$40$	0	0
$41$	3.19208	0.498519	0.249259	−	0.968437i	$-0.419813\pi$
$41$	3.19208	0.498519	0.249259	+	0.968437i	$0.419813\pi$
$42$	0	0
$43$	4.90755	0.748394	0.374197	−	0.927349i	$-0.377918\pi$
$43$	4.90755	0.748394	0.374197	+	0.927349i	$0.377918\pi$
$44$	−2.39327	−0.360799
$45$	0	0
$46$	8.17113	1.20477
$47$	−3.00868	−0.438860	−0.219430	−	0.975628i	$-0.570420\pi$
$47$	−3.00868	−0.438860	−0.219430	+	0.975628i	$0.570420\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	0	0
$51$	1.97038	0.275909
$52$	−3.80748	−0.528003
$53$	12.1711	1.67183	0.835917	−	0.548856i	$-0.184936\pi$
$53$	12.1711	1.67183	0.835917	+	0.548856i	$0.184936\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	−4.17113	−0.552480
$58$	−9.16246	−1.20309
$59$	−13.0414	−1.69785	−0.848926	−	0.528512i	$-0.822750\pi$
$59$	−13.0414	−1.69785	−0.848926	+	0.528512i	$0.822750\pi$
$60$	0	0
$61$	−13.0910	−1.67612	−0.838062	−	0.545574i	$-0.816312\pi$
$61$	−13.0910	−1.67612	−0.838062	+	0.545574i	$0.816312\pi$
$62$	−1.60673	−0.204055
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	2.39327	0.294591
$67$	−5.77786	−0.705878	−0.352939	−	0.935646i	$-0.614818\pi$
$67$	−5.77786	−0.705878	−0.352939	+	0.935646i	$0.614818\pi$
$68$	−1.97038	−0.238944
$69$	−8.17113	−0.983689
$70$	0	0
$71$	6.94032	0.823665	0.411832	−	0.911260i	$-0.364889\pi$
$71$	6.94032	0.823665	0.411832	+	0.911260i	$0.364889\pi$
$72$	1.00000	0.117851
$73$	0.506664	0.0593005	0.0296503	−	0.999560i	$-0.490561\pi$
$73$	0.506664	0.0593005	0.0296503	+	0.999560i	$0.490561\pi$
$74$	1.26358	0.146889
$75$	0	0
$76$	4.17113	0.478462
$77$	0	0
$78$	3.80748	0.431113
$79$	−12.4260	−1.39804	−0.699020	−	0.715103i	$-0.746380\pi$
$79$	−12.4260	−1.39804	−0.699020	+	0.715103i	$0.746380\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.19208	0.352506
$83$	1.89887	0.208429	0.104214	−	0.994555i	$-0.466767\pi$
$83$	1.89887	0.208429	0.104214	+	0.994555i	$0.466767\pi$
$84$	0	0
$85$	0	0
$86$	4.90755	0.529195
$87$	9.16246	0.982319
$88$	−2.39327	−0.255123
$89$	−1.97949	−0.209826	−0.104913	−	0.994481i	$-0.533456\pi$
$89$	−1.97949	−0.209826	−0.104913	+	0.994481i	$0.533456\pi$
$90$	0	0
$91$	0	0
$92$	8.17113	0.851900
$93$	1.60673	0.166610
$94$	−3.00868	−0.310321
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−12.0624	−1.22475	−0.612375	−	0.790567i	$-0.709786\pi$
$97$	−12.0624	−1.22475	−0.612375	+	0.790567i	$0.709786\pi$
$98$	0	0
$99$	−2.39327	−0.240533
$100$	0	0
$101$	−7.86672	−0.782768	−0.391384	−	0.920227i	$-0.628004\pi$
$101$	−7.86672	−0.782768	−0.391384	+	0.920227i	$0.628004\pi$
$102$	1.97038	0.195097
$103$	−5.89887	−0.581233	−0.290617	−	0.956840i	$-0.593860\pi$
$103$	−5.89887	−0.581233	−0.290617	+	0.956840i	$0.593860\pi$
$104$	−3.80748	−0.373354
$105$	0	0
$106$	12.1711	1.18217
$107$	3.75798	0.363298	0.181649	−	0.983363i	$-0.441857\pi$
$107$	3.75798	0.363298	0.181649	+	0.983363i	$0.441857\pi$
$108$	−1.00000	−0.0962250
$109$	7.14257	0.684135	0.342067	−	0.939675i	$-0.388873\pi$
$109$	7.14257	0.684135	0.342067	+	0.939675i	$0.388873\pi$
$110$	0	0
$111$	−1.26358	−0.119934
$112$	0	0
$113$	−17.3837	−1.63532	−0.817661	−	0.575700i	$-0.804730\pi$
$113$	−17.3837	−1.63532	−0.817661	+	0.575700i	$0.804730\pi$
$114$	−4.17113	−0.390663
$115$	0	0
$116$	−9.16246	−0.850713
$117$	−3.80748	−0.352002
$118$	−13.0414	−1.20056
$119$	0	0
$120$	0	0
$121$	−5.27226	−0.479296
$122$	−13.0910	−1.18520
$123$	−3.19208	−0.287820
$124$	−1.60673	−0.144289
$125$	0	0
$126$	0	0
$127$	−18.4434	−1.63659	−0.818293	−	0.574801i	$-0.805079\pi$
$127$	−18.4434	−1.63659	−0.818293	+	0.574801i	$0.805079\pi$
$128$	1.00000	0.0883883
$129$	−4.90755	−0.432086
$130$	0	0
$131$	−0.615405	−0.0537682	−0.0268841	−	0.999639i	$-0.508559\pi$
$131$	−0.615405	−0.0537682	−0.0268841	+	0.999639i	$0.508559\pi$
$132$	2.39327	0.208307
$133$	0	0
$134$	−5.77786	−0.499131
$135$	0	0
$136$	−1.97038	−0.168959
$137$	−7.82799	−0.668790	−0.334395	−	0.942433i	$-0.608532\pi$
$137$	−7.82799	−0.668790	−0.334395	+	0.942433i	$0.608532\pi$
$138$	−8.17113	−0.695573
$139$	6.15378	0.521957	0.260979	−	0.965345i	$-0.415955\pi$
$139$	6.15378	0.521957	0.260979	+	0.965345i	$0.415955\pi$
$140$	0	0
$141$	3.00868	0.253376
$142$	6.94032	0.582419
$143$	9.11233	0.762012
$144$	1.00000	0.0833333
$145$	0	0
$146$	0.506664	0.0419318
$147$	0	0
$148$	1.26358	0.103866
$149$	−7.53584	−0.617360	−0.308680	−	0.951166i	$-0.599887\pi$
$149$	−7.53584	−0.617360	−0.308680	+	0.951166i	$0.599887\pi$
$150$	0	0
$151$	9.04145	0.735783	0.367891	−	0.929869i	$-0.380080\pi$
$151$	9.04145	0.735783	0.367891	+	0.929869i	$0.380080\pi$
$152$	4.17113	0.338324
$153$	−1.97038	−0.159296
$154$	0	0
$155$	0	0
$156$	3.80748	0.304843
$157$	22.6640	1.80879	0.904393	−	0.426700i	$-0.140324\pi$
$157$	22.6640	1.80879	0.904393	+	0.426700i	$0.140324\pi$
$158$	−12.4260	−0.988563
$159$	−12.1711	−0.965234
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	10.2039	0.799232	0.399616	−	0.916683i	$-0.369144\pi$
$163$	10.2039	0.799232	0.399616	+	0.916683i	$0.369144\pi$
$164$	3.19208	0.249259
$165$	0	0
$166$	1.89887	0.147381
$167$	−25.2498	−1.95389	−0.976945	−	0.213492i	$-0.931516\pi$
$167$	−25.2498	−1.95389	−0.976945	+	0.213492i	$0.931516\pi$
$168$	0	0
$169$	1.49693	0.115148
$170$	0	0
$171$	4.17113	0.318975
$172$	4.90755	0.374197
$173$	12.5348	0.953002	0.476501	−	0.879174i	$-0.341905\pi$
$173$	12.5348	0.953002	0.476501	+	0.879174i	$0.341905\pi$
$174$	9.16246	0.694604
$175$	0	0
$176$	−2.39327	−0.180399
$177$	13.0414	0.980255
$178$	−1.97949	−0.148369
$179$	22.9196	1.71309	0.856544	−	0.516074i	$-0.172607\pi$
$179$	22.9196	1.71309	0.856544	+	0.516074i	$0.172607\pi$
$180$	0	0
$181$	−19.6181	−1.45820	−0.729102	−	0.684405i	$-0.760062\pi$
$181$	−19.6181	−1.45820	−0.729102	+	0.684405i	$0.760062\pi$
$182$	0	0
$183$	13.0910	0.967711
$184$	8.17113	0.602384
$185$	0	0
$186$	1.60673	0.117811
$187$	4.71565	0.344843
$188$	−3.00868	−0.219430
$189$	0	0
$190$	0	0
$191$	−19.7269	−1.42739	−0.713693	−	0.700459i	$-0.752979\pi$
$191$	−19.7269	−1.42739	−0.713693	+	0.700459i	$0.752979\pi$
$192$	−1.00000	−0.0721688
$193$	−4.78654	−0.344543	−0.172271	−	0.985050i	$-0.555111\pi$
$193$	−4.78654	−0.344543	−0.172271	+	0.985050i	$0.555111\pi$
$194$	−12.0624	−0.866029
$195$	0	0
$196$	0	0
$197$	17.1997	1.22543	0.612714	−	0.790305i	$-0.290078\pi$
$197$	17.1997	1.22543	0.612714	+	0.790305i	$0.290078\pi$
$198$	−2.39327	−0.170082
$199$	−15.5056	−1.09916	−0.549582	−	0.835440i	$-0.685213\pi$
$199$	−15.5056	−1.09916	−0.549582	+	0.835440i	$0.685213\pi$
$200$	0	0
$201$	5.77786	0.407539
$202$	−7.86672	−0.553501
$203$	0	0
$204$	1.97038	0.137954
$205$	0	0
$206$	−5.89887	−0.410994
$207$	8.17113	0.567933
$208$	−3.80748	−0.264001
$209$	−9.98265	−0.690514
$210$	0	0
$211$	−23.8980	−1.64521	−0.822603	−	0.568616i	$-0.807479\pi$
$211$	−23.8980	−1.64521	−0.822603	+	0.568616i	$0.807479\pi$
$212$	12.1711	0.835917
$213$	−6.94032	−0.475543
$214$	3.75798	0.256890
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	7.14257	0.483756
$219$	−0.506664	−0.0342372
$220$	0	0
$221$	7.50219	0.504652
$222$	−1.26358	−0.0848062
$223$	−0.870315	−0.0582806	−0.0291403	−	0.999575i	$-0.509277\pi$
$223$	−0.870315	−0.0582806	−0.0291403	+	0.999575i	$0.509277\pi$
$224$	0	0
$225$	0	0
$226$	−17.3837	−1.15635
$227$	−14.6854	−0.974705	−0.487353	−	0.873205i	$-0.662037\pi$
$227$	−14.6854	−0.974705	−0.487353	+	0.873205i	$0.662037\pi$
$228$	−4.17113	−0.276240
$229$	−24.0379	−1.58847	−0.794233	−	0.607613i	$-0.792127\pi$
$229$	−24.0379	−1.58847	−0.794233	+	0.607613i	$0.792127\pi$
$230$	0	0
$231$	0	0
$232$	−9.16246	−0.601545
$233$	10.5445	0.690794	0.345397	−	0.938457i	$-0.387744\pi$
$233$	10.5445	0.690794	0.345397	+	0.938457i	$0.387744\pi$
$234$	−3.80748	−0.248903
$235$	0	0
$236$	−13.0414	−0.848926
$237$	12.4260	0.807158
$238$	0	0
$239$	−4.01289	−0.259572	−0.129786	−	0.991542i	$-0.541429\pi$
$239$	−4.01289	−0.259572	−0.129786	+	0.991542i	$0.541429\pi$
$240$	0	0
$241$	−0.543899	−0.0350356	−0.0175178	−	0.999847i	$-0.505576\pi$
$241$	−0.543899	−0.0350356	−0.0175178	+	0.999847i	$0.505576\pi$
$242$	−5.27226	−0.338914
$243$	−1.00000	−0.0641500
$244$	−13.0910	−0.838062
$245$	0	0
$246$	−3.19208	−0.203519
$247$	−15.8815	−1.01052
$248$	−1.60673	−0.102027
$249$	−1.89887	−0.120336
$250$	0	0
$251$	4.62829	0.292135	0.146068	−	0.989275i	$-0.453338\pi$
$251$	4.62829	0.292135	0.146068	+	0.989275i	$0.453338\pi$
$252$	0	0
$253$	−19.5557	−1.22946
$254$	−18.4434	−1.15724
$255$	0	0
$256$	1.00000	0.0625000
$257$	−25.5671	−1.59483	−0.797417	−	0.603429i	$-0.793801\pi$
$257$	−25.5671	−1.59483	−0.797417	+	0.603429i	$0.793801\pi$
$258$	−4.90755	−0.305531
$259$	0	0
$260$	0	0
$261$	−9.16246	−0.567142
$262$	−0.615405	−0.0380199
$263$	20.5972	1.27008	0.635038	−	0.772481i	$-0.280984\pi$
$263$	20.5972	1.27008	0.635038	+	0.772481i	$0.280984\pi$
$264$	2.39327	0.147296
$265$	0	0
$266$	0	0
$267$	1.97949	0.121143
$268$	−5.77786	−0.352939
$269$	2.23799	0.136453	0.0682263	−	0.997670i	$-0.478266\pi$
$269$	2.23799	0.136453	0.0682263	+	0.997670i	$0.478266\pi$
$270$	0	0
$271$	19.1616	1.16398	0.581992	−	0.813195i	$-0.302274\pi$
$271$	19.1616	1.16398	0.581992	+	0.813195i	$0.302274\pi$
$272$	−1.97038	−0.119472
$273$	0	0
$274$	−7.82799	−0.472906
$275$	0	0
$276$	−8.17113	−0.491844
$277$	3.67674	0.220914	0.110457	−	0.993881i	$-0.464769\pi$
$277$	3.67674	0.220914	0.110457	+	0.993881i	$0.464769\pi$
$278$	6.15378	0.369079
$279$	−1.60673	−0.0961924
$280$	0	0
$281$	−29.8280	−1.77939	−0.889694	−	0.456557i	$-0.849083\pi$
$281$	−29.8280	−1.77939	−0.889694	+	0.456557i	$0.849083\pi$
$282$	3.00868	0.179164
$283$	4.28961	0.254991	0.127495	−	0.991839i	$-0.459306\pi$
$283$	4.28961	0.254991	0.127495	+	0.991839i	$0.459306\pi$
$284$	6.94032	0.411832
$285$	0	0
$286$	9.11233	0.538824
$287$	0	0
$288$	1.00000	0.0589256
$289$	−13.1176	−0.771623
$290$	0	0
$291$	12.0624	0.707110
$292$	0.506664	0.0296503
$293$	24.5940	1.43680	0.718399	−	0.695631i	$-0.244875\pi$
$293$	24.5940	1.43680	0.718399	+	0.695631i	$0.244875\pi$
$294$	0	0
$295$	0	0
$296$	1.26358	0.0734444
$297$	2.39327	0.138872
$298$	−7.53584	−0.436540
$299$	−31.1115	−1.79922
$300$	0	0
$301$	0	0
$302$	9.04145	0.520277
$303$	7.86672	0.451931
$304$	4.17113	0.239231
$305$	0	0
$306$	−1.97038	−0.112639
$307$	2.14089	0.122187	0.0610937	−	0.998132i	$-0.480541\pi$
$307$	2.14089	0.122187	0.0610937	+	0.998132i	$0.480541\pi$
$308$	0	0
$309$	5.89887	0.335575
$310$	0	0
$311$	−13.9991	−0.793817	−0.396909	−	0.917858i	$-0.629917\pi$
$311$	−13.9991	−0.793817	−0.396909	+	0.917858i	$0.629917\pi$
$312$	3.80748	0.215556
$313$	33.0727	1.86938	0.934690	−	0.355463i	$-0.115677\pi$
$313$	33.0727	1.86938	0.934690	+	0.355463i	$0.115677\pi$
$314$	22.6640	1.27901
$315$	0	0
$316$	−12.4260	−0.699020
$317$	7.38459	0.414760	0.207380	−	0.978260i	$-0.433506\pi$
$317$	7.38459	0.414760	0.207380	+	0.978260i	$0.433506\pi$
$318$	−12.1711	−0.682523
$319$	21.9282	1.22775
$320$	0	0
$321$	−3.75798	−0.209750
$322$	0	0
$323$	−8.21872	−0.457302
$324$	1.00000	0.0555556
$325$	0	0
$326$	10.2039	0.565142
$327$	−7.14257	−0.394985
$328$	3.19208	0.176253
$329$	0	0
$330$	0	0
$331$	21.5549	1.18476	0.592381	−	0.805658i	$-0.298188\pi$
$331$	21.5549	1.18476	0.592381	+	0.805658i	$0.298188\pi$
$332$	1.89887	0.104214
$333$	1.26358	0.0692440
$334$	−25.2498	−1.38161
$335$	0	0
$336$	0	0
$337$	−22.2549	−1.21230	−0.606151	−	0.795350i	$-0.707287\pi$
$337$	−22.2549	−1.21230	−0.606151	+	0.795350i	$0.707287\pi$
$338$	1.49693	0.0814222
$339$	17.3837	0.944154
$340$	0	0
$341$	3.84534	0.208237
$342$	4.17113	0.225549
$343$	0	0
$344$	4.90755	0.264597
$345$	0	0
$346$	12.5348	0.673874
$347$	−2.14089	−0.114929	−0.0574646	−	0.998348i	$-0.518302\pi$
$347$	−2.14089	−0.114929	−0.0574646	+	0.998348i	$0.518302\pi$
$348$	9.16246	0.491159
$349$	0.906929	0.0485468	0.0242734	−	0.999705i	$-0.492273\pi$
$349$	0.906929	0.0485468	0.0242734	+	0.999705i	$0.492273\pi$
$350$	0	0
$351$	3.80748	0.203228
$352$	−2.39327	−0.127562
$353$	13.8866	0.739109	0.369555	−	0.929209i	$-0.379510\pi$
$353$	13.8866	0.739109	0.369555	+	0.929209i	$0.379510\pi$
$354$	13.0414	0.693145
$355$	0	0
$356$	−1.97949	−0.104913
$357$	0	0
$358$	22.9196	1.21134
$359$	18.8565	0.995211	0.497605	−	0.867404i	$-0.334213\pi$
$359$	18.8565	0.995211	0.497605	+	0.867404i	$0.334213\pi$
$360$	0	0
$361$	−1.60164	−0.0842968
$362$	−19.6181	−1.03111
$363$	5.27226	0.276722
$364$	0	0
$365$	0	0
$366$	13.0910	0.684275
$367$	−31.5722	−1.64806	−0.824028	−	0.566549i	$-0.808278\pi$
$367$	−31.5722	−1.64806	−0.824028	+	0.566549i	$0.808278\pi$
$368$	8.17113	0.425950
$369$	3.19208	0.166173
$370$	0	0
$371$	0	0
$372$	1.60673	0.0833051
$373$	−20.9075	−1.08255	−0.541276	−	0.840845i	$-0.682059\pi$
$373$	−20.9075	−1.08255	−0.541276	+	0.840845i	$0.682059\pi$
$374$	4.71565	0.243841
$375$	0	0
$376$	−3.00868	−0.155161
$377$	34.8859	1.79672
$378$	0	0
$379$	−9.82440	−0.504646	−0.252323	−	0.967643i	$-0.581195\pi$
$379$	−9.82440	−0.504646	−0.252323	+	0.967643i	$0.581195\pi$
$380$	0	0
$381$	18.4434	0.944884
$382$	−19.7269	−1.00931
$383$	−30.6647	−1.56689	−0.783445	−	0.621461i	$-0.786540\pi$
$383$	−30.6647	−1.56689	−0.783445	+	0.621461i	$0.786540\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−4.78654	−0.243628
$387$	4.90755	0.249465
$388$	−12.0624	−0.612375
$389$	23.4476	1.18884	0.594420	−	0.804154i	$-0.297382\pi$
$389$	23.4476	1.18884	0.594420	+	0.804154i	$0.297382\pi$
$390$	0	0
$391$	−16.1002	−0.814225
$392$	0	0
$393$	0.615405	0.0310431
$394$	17.1997	0.866508
$395$	0	0
$396$	−2.39327	−0.120266
$397$	35.3761	1.77548	0.887738	−	0.460349i	$-0.152276\pi$
$397$	35.3761	1.77548	0.887738	+	0.460349i	$0.152276\pi$
$398$	−15.5056	−0.777226
$399$	0	0
$400$	0	0
$401$	−9.04145	−0.451508	−0.225754	−	0.974184i	$-0.572485\pi$
$401$	−9.04145	−0.451508	−0.225754	+	0.974184i	$0.572485\pi$
$402$	5.77786	0.288174
$403$	6.11760	0.304739
$404$	−7.86672	−0.391384
$405$	0	0
$406$	0	0
$407$	−3.02410	−0.149899
$408$	1.97038	0.0975484
$409$	15.4379	0.763354	0.381677	−	0.924296i	$-0.375347\pi$
$409$	15.4379	0.763354	0.381677	+	0.924296i	$0.375347\pi$
$410$	0	0
$411$	7.82799	0.386126
$412$	−5.89887	−0.290617
$413$	0	0
$414$	8.17113	0.401589
$415$	0	0
$416$	−3.80748	−0.186677
$417$	−6.15378	−0.301352
$418$	−9.98265	−0.488267
$419$	26.6274	1.30083	0.650417	−	0.759577i	$-0.274594\pi$
$419$	26.6274	1.30083	0.650417	+	0.759577i	$0.274594\pi$
$420$	0	0
$421$	5.04591	0.245923	0.122961	−	0.992411i	$-0.460761\pi$
$421$	5.04591	0.245923	0.122961	+	0.992411i	$0.460761\pi$
$422$	−23.8980	−1.16334
$423$	−3.00868	−0.146287
$424$	12.1711	0.591083
$425$	0	0
$426$	−6.94032	−0.336260
$427$	0	0
$428$	3.75798	0.181649
$429$	−9.11233	−0.439948
$430$	0	0
$431$	−25.2835	−1.21786	−0.608931	−	0.793223i	$-0.708401\pi$
$431$	−25.2835	−1.21786	−0.608931	+	0.793223i	$0.708401\pi$
$432$	−1.00000	−0.0481125
$433$	−27.0063	−1.29784	−0.648920	−	0.760857i	$-0.724779\pi$
$433$	−27.0063	−1.29784	−0.648920	+	0.760857i	$0.724779\pi$
$434$	0	0
$435$	0	0
$436$	7.14257	0.342067
$437$	34.0829	1.63041
$438$	−0.506664	−0.0242093
$439$	−27.9316	−1.33310	−0.666552	−	0.745458i	$-0.732231\pi$
$439$	−27.9316	−1.33310	−0.666552	+	0.745458i	$0.732231\pi$
$440$	0	0
$441$	0	0
$442$	7.50219	0.356843
$443$	10.7692	0.511660	0.255830	−	0.966722i	$-0.417651\pi$
$443$	10.7692	0.511660	0.255830	+	0.966722i	$0.417651\pi$
$444$	−1.26358	−0.0599671
$445$	0	0
$446$	−0.870315	−0.0412106
$447$	7.53584	0.356433
$448$	0	0
$449$	−15.1288	−0.713973	−0.356986	−	0.934110i	$-0.616196\pi$
$449$	−15.1288	−0.713973	−0.356986	+	0.934110i	$0.616196\pi$
$450$	0	0
$451$	−7.63950	−0.359730
$452$	−17.3837	−0.817661
$453$	−9.04145	−0.424804
$454$	−14.6854	−0.689221
$455$	0	0
$456$	−4.17113	−0.195331
$457$	21.4711	1.00437	0.502187	−	0.864759i	$-0.332529\pi$
$457$	21.4711	1.00437	0.502187	+	0.864759i	$0.332529\pi$
$458$	−24.0379	−1.12322
$459$	1.97038	0.0919695
$460$	0	0
$461$	30.4465	1.41804	0.709019	−	0.705190i	$-0.249138\pi$
$461$	30.4465	1.41804	0.709019	+	0.705190i	$0.249138\pi$
$462$	0	0
$463$	−40.2005	−1.86828	−0.934138	−	0.356913i	$-0.883829\pi$
$463$	−40.2005	−1.86828	−0.934138	+	0.356913i	$0.883829\pi$
$464$	−9.16246	−0.425356
$465$	0	0
$466$	10.5445	0.488465
$467$	−7.84175	−0.362873	−0.181437	−	0.983403i	$-0.558075\pi$
$467$	−7.84175	−0.362873	−0.181437	+	0.983403i	$0.558075\pi$
$468$	−3.80748	−0.176001
$469$	0	0
$470$	0	0
$471$	−22.6640	−1.04430
$472$	−13.0414	−0.600281
$473$	−11.7451	−0.540040
$474$	12.4260	0.570747
$475$	0	0
$476$	0	0
$477$	12.1711	0.557278
$478$	−4.01289	−0.183545
$479$	−27.7745	−1.26905	−0.634524	−	0.772904i	$-0.718804\pi$
$479$	−27.7745	−1.26905	−0.634524	+	0.772904i	$0.718804\pi$
$480$	0	0
$481$	−4.81108	−0.219366
$482$	−0.543899	−0.0247739
$483$	0	0
$484$	−5.27226	−0.239648
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	12.5263	0.567620	0.283810	−	0.958880i	$-0.408401\pi$
$487$	12.5263	0.567620	0.283810	+	0.958880i	$0.408401\pi$
$488$	−13.0910	−0.592600
$489$	−10.2039	−0.461437
$490$	0	0
$491$	−24.3924	−1.10081	−0.550407	−	0.834897i	$-0.685527\pi$
$491$	−24.3924	−1.10081	−0.550407	+	0.834897i	$0.685527\pi$
$492$	−3.19208	−0.143910
$493$	18.0535	0.813090
$494$	−15.8815	−0.714544
$495$	0	0
$496$	−1.60673	−0.0721443
$497$	0	0
$498$	−1.89887	−0.0850906
$499$	−23.6560	−1.05899	−0.529493	−	0.848314i	$-0.677618\pi$
$499$	−23.6560	−1.05899	−0.529493	+	0.848314i	$0.677618\pi$
$500$	0	0
$501$	25.2498	1.12808
$502$	4.62829	0.206571
$503$	20.8496	0.929636	0.464818	−	0.885406i	$-0.346120\pi$
$503$	20.8496	0.929636	0.464818	+	0.885406i	$0.346120\pi$
$504$	0	0
$505$	0	0
$506$	−19.5557	−0.869358
$507$	−1.49693	−0.0664810
$508$	−18.4434	−0.818293
$509$	−2.59490	−0.115017	−0.0575085	−	0.998345i	$-0.518316\pi$
$509$	−2.59490	−0.115017	−0.0575085	+	0.998345i	$0.518316\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	−4.17113	−0.184160
$514$	−25.5671	−1.12772
$515$	0	0
$516$	−4.90755	−0.216043
$517$	7.20057	0.316681
$518$	0	0
$519$	−12.5348	−0.550216
$520$	0	0
$521$	−24.4884	−1.07286	−0.536429	−	0.843945i	$-0.680227\pi$
$521$	−24.4884	−1.07286	−0.536429	+	0.843945i	$0.680227\pi$
$522$	−9.16246	−0.401030
$523$	−21.6257	−0.945627	−0.472814	−	0.881162i	$-0.656762\pi$
$523$	−21.6257	−0.945627	−0.472814	+	0.881162i	$0.656762\pi$
$524$	−0.615405	−0.0268841
$525$	0	0
$526$	20.5972	0.898080
$527$	3.16587	0.137907
$528$	2.39327	0.104154
$529$	43.7674	1.90293
$530$	0	0
$531$	−13.0414	−0.565951
$532$	0	0
$533$	−12.1538	−0.526439
$534$	1.97949	0.0856611
$535$	0	0
$536$	−5.77786	−0.249566
$537$	−22.9196	−0.989052
$538$	2.23799	0.0964865
$539$	0	0
$540$	0	0
$541$	8.01735	0.344693	0.172346	−	0.985036i	$-0.444865\pi$
$541$	8.01735	0.344693	0.172346	+	0.985036i	$0.444865\pi$
$542$	19.1616	0.823060
$543$	19.6181	0.841894
$544$	−1.97038	−0.0844794
$545$	0	0
$546$	0	0
$547$	−43.0251	−1.83962	−0.919811	−	0.392361i	$-0.871658\pi$
$547$	−43.0251	−1.83962	−0.919811	+	0.392361i	$0.871658\pi$
$548$	−7.82799	−0.334395
$549$	−13.0910	−0.558708
$550$	0	0
$551$	−38.2178	−1.62814
$552$	−8.17113	−0.347787
$553$	0	0
$554$	3.67674	0.156210
$555$	0	0
$556$	6.15378	0.260979
$557$	11.1823	0.473811	0.236906	−	0.971533i	$-0.423867\pi$
$557$	11.1823	0.473811	0.236906	+	0.971533i	$0.423867\pi$
$558$	−1.60673	−0.0680183
$559$	−18.6854	−0.790309
$560$	0	0
$561$	−4.71565	−0.199095
$562$	−29.8280	−1.25822
$563$	26.8694	1.13241	0.566206	−	0.824264i	$-0.308411\pi$
$563$	26.8694	1.13241	0.566206	+	0.824264i	$0.308411\pi$
$564$	3.00868	0.126688
$565$	0	0
$566$	4.28961	0.180306
$567$	0	0
$568$	6.94032	0.291210
$569$	−2.61453	−0.109607	−0.0548034	−	0.998497i	$-0.517453\pi$
$569$	−2.61453	−0.109607	−0.0548034	+	0.998497i	$0.517453\pi$
$570$	0	0
$571$	12.4607	0.521466	0.260733	−	0.965411i	$-0.416036\pi$
$571$	12.4607	0.521466	0.260733	+	0.965411i	$0.416036\pi$
$572$	9.11233	0.381006
$573$	19.7269	0.824102
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−17.8195	−0.741835	−0.370918	−	0.928666i	$-0.620957\pi$
$577$	−17.8195	−0.741835	−0.370918	+	0.928666i	$0.620957\pi$
$578$	−13.1176	−0.545620
$579$	4.78654	0.198922
$580$	0	0
$581$	0	0
$582$	12.0624	0.500002
$583$	−29.1288	−1.20639
$584$	0.506664	0.0209659
$585$	0	0
$586$	24.5940	1.01597
$587$	9.55661	0.394443	0.197222	−	0.980359i	$-0.436808\pi$
$587$	9.55661	0.394443	0.197222	+	0.980359i	$0.436808\pi$
$588$	0	0
$589$	−6.70189	−0.276146
$590$	0	0
$591$	−17.1997	−0.707501
$592$	1.26358	0.0519330
$593$	11.6619	0.478898	0.239449	−	0.970909i	$-0.423033\pi$
$593$	11.6619	0.478898	0.239449	+	0.970909i	$0.423033\pi$
$594$	2.39327	0.0981970
$595$	0	0
$596$	−7.53584	−0.308680
$597$	15.5056	0.634602
$598$	−31.1115	−1.27224
$599$	−33.0977	−1.35234	−0.676168	−	0.736748i	$-0.736360\pi$
$599$	−33.0977	−1.35234	−0.676168	+	0.736748i	$0.736360\pi$
$600$	0	0
$601$	−15.2886	−0.623633	−0.311817	−	0.950142i	$-0.600937\pi$
$601$	−15.2886	−0.623633	−0.311817	+	0.950142i	$0.600937\pi$
$602$	0	0
$603$	−5.77786	−0.235293
$604$	9.04145	0.367891
$605$	0	0
$606$	7.86672	0.319564
$607$	40.1823	1.63095	0.815474	−	0.578794i	$-0.196476\pi$
$607$	40.1823	1.63095	0.815474	+	0.578794i	$0.196476\pi$
$608$	4.17113	0.169162
$609$	0	0
$610$	0	0
$611$	11.4555	0.463439
$612$	−1.97038	−0.0796479
$613$	−24.8591	−1.00405	−0.502024	−	0.864853i	$-0.667411\pi$
$613$	−24.8591	−1.00405	−0.502024	+	0.864853i	$0.667411\pi$
$614$	2.14089	0.0863995
$615$	0	0
$616$	0	0
$617$	22.7647	0.916473	0.458237	−	0.888830i	$-0.348481\pi$
$617$	22.7647	0.916473	0.458237	+	0.888830i	$0.348481\pi$
$618$	5.89887	0.237288
$619$	−17.8280	−0.716567	−0.358284	−	0.933613i	$-0.616638\pi$
$619$	−17.8280	−0.716567	−0.358284	+	0.933613i	$0.616638\pi$
$620$	0	0
$621$	−8.17113	−0.327896
$622$	−13.9991	−0.561314
$623$	0	0
$624$	3.80748	0.152421
$625$	0	0
$626$	33.0727	1.32185
$627$	9.98265	0.398669
$628$	22.6640	0.904393
$629$	−2.48974	−0.0992725
$630$	0	0
$631$	0.580705	0.0231175	0.0115587	−	0.999933i	$-0.496321\pi$
$631$	0.580705	0.0231175	0.0115587	+	0.999933i	$0.496321\pi$
$632$	−12.4260	−0.494281
$633$	23.8980	0.949860
$634$	7.38459	0.293280
$635$	0	0
$636$	−12.1711	−0.482617
$637$	0	0
$638$	21.9282	0.868147
$639$	6.94032	0.274555
$640$	0	0
$641$	18.0353	0.712352	0.356176	−	0.934419i	$-0.384080\pi$
$641$	18.0353	0.712352	0.356176	+	0.934419i	$0.384080\pi$
$642$	−3.75798	−0.148316
$643$	39.3664	1.55246	0.776229	−	0.630451i	$-0.217130\pi$
$643$	39.3664	1.55246	0.776229	+	0.630451i	$0.217130\pi$
$644$	0	0
$645$	0	0
$646$	−8.21872	−0.323361
$647$	2.38038	0.0935824	0.0467912	−	0.998905i	$-0.485100\pi$
$647$	2.38038	0.0935824	0.0467912	+	0.998905i	$0.485100\pi$
$648$	1.00000	0.0392837
$649$	31.2117	1.22517
$650$	0	0
$651$	0	0
$652$	10.2039	0.399616
$653$	−25.1115	−0.982687	−0.491344	−	0.870966i	$-0.663494\pi$
$653$	−25.1115	−0.982687	−0.491344	+	0.870966i	$0.663494\pi$
$654$	−7.14257	−0.279297
$655$	0	0
$656$	3.19208	0.124630
$657$	0.506664	0.0197668
$658$	0	0
$659$	−7.17981	−0.279686	−0.139843	−	0.990174i	$-0.544660\pi$
$659$	−7.17981	−0.279686	−0.139843	+	0.990174i	$0.544660\pi$
$660$	0	0
$661$	25.0330	0.973669	0.486835	−	0.873494i	$-0.338151\pi$
$661$	25.0330	0.973669	0.486835	+	0.873494i	$0.338151\pi$
$662$	21.5549	0.837753
$663$	−7.50219	−0.291361
$664$	1.89887	0.0736906
$665$	0	0
$666$	1.26358	0.0489629
$667$	−74.8677	−2.89889
$668$	−25.2498	−0.976945
$669$	0.870315	0.0336483
$670$	0	0
$671$	31.3302	1.20949
$672$	0	0
$673$	−9.67062	−0.372775	−0.186388	−	0.982476i	$-0.559678\pi$
$673$	−9.67062	−0.372775	−0.186388	+	0.982476i	$0.559678\pi$
$674$	−22.2549	−0.857227
$675$	0	0
$676$	1.49693	0.0575742
$677$	27.6890	1.06418	0.532088	−	0.846689i	$-0.321408\pi$
$677$	27.6890	1.06418	0.532088	+	0.846689i	$0.321408\pi$
$678$	17.3837	0.667618
$679$	0	0
$680$	0	0
$681$	14.6854	0.562746
$682$	3.84534	0.147246
$683$	−26.1227	−0.999556	−0.499778	−	0.866154i	$-0.666585\pi$
$683$	−26.1227	−0.999556	−0.499778	+	0.866154i	$0.666585\pi$
$684$	4.17113	0.159487
$685$	0	0
$686$	0	0
$687$	24.0379	0.917101
$688$	4.90755	0.187099
$689$	−46.3414	−1.76547
$690$	0	0
$691$	−24.7901	−0.943061	−0.471530	−	0.881850i	$-0.656298\pi$
$691$	−24.7901	−0.943061	−0.471530	+	0.881850i	$0.656298\pi$
$692$	12.5348	0.476501
$693$	0	0
$694$	−2.14089	−0.0812673
$695$	0	0
$696$	9.16246	0.347302
$697$	−6.28961	−0.238236
$698$	0.906929	0.0343278
$699$	−10.5445	−0.398830
$700$	0	0
$701$	37.8817	1.43077	0.715386	−	0.698730i	$-0.246251\pi$
$701$	37.8817	1.43077	0.715386	+	0.698730i	$0.246251\pi$
$702$	3.80748	0.143704
$703$	5.27058	0.198784
$704$	−2.39327	−0.0901997
$705$	0	0
$706$	13.8866	0.522629
$707$	0	0
$708$	13.0414	0.490128
$709$	−30.9801	−1.16348	−0.581741	−	0.813374i	$-0.697628\pi$
$709$	−30.9801	−1.16348	−0.581741	+	0.813374i	$0.697628\pi$
$710$	0	0
$711$	−12.4260	−0.466013
$712$	−1.97949	−0.0741847
$713$	−13.1288	−0.491678
$714$	0	0
$715$	0	0
$716$	22.9196	0.856544
$717$	4.01289	0.149864
$718$	18.8565	0.703720
$719$	16.7028	0.622908	0.311454	−	0.950261i	$-0.399184\pi$
$719$	16.7028	0.622908	0.311454	+	0.950261i	$0.399184\pi$
$720$	0	0
$721$	0	0
$722$	−1.60164	−0.0596068
$723$	0.543899	0.0202278
$724$	−19.6181	−0.729102
$725$	0	0
$726$	5.27226	0.195672
$727$	−36.1218	−1.33968	−0.669842	−	0.742504i	$-0.733638\pi$
$727$	−36.1218	−1.33968	−0.669842	+	0.742504i	$0.733638\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.66974	−0.357648
$732$	13.0910	0.483856
$733$	7.83860	0.289525	0.144763	−	0.989466i	$-0.453758\pi$
$733$	7.83860	0.289525	0.144763	+	0.989466i	$0.453758\pi$
$734$	−31.5722	−1.16535
$735$	0	0
$736$	8.17113	0.301192
$737$	13.8280	0.509361
$738$	3.19208	0.117502
$739$	−29.7969	−1.09610	−0.548048	−	0.836447i	$-0.684629\pi$
$739$	−29.7969	−1.09610	−0.548048	+	0.836447i	$0.684629\pi$
$740$	0	0
$741$	15.8815	0.583422
$742$	0	0
$743$	11.7487	0.431017	0.215509	−	0.976502i	$-0.430859\pi$
$743$	11.7487	0.431017	0.215509	+	0.976502i	$0.430859\pi$
$744$	1.60673	0.0589056
$745$	0	0
$746$	−20.9075	−0.765480
$747$	1.89887	0.0694762
$748$	4.71565	0.172421
$749$	0	0
$750$	0	0
$751$	−2.43981	−0.0890299	−0.0445150	−	0.999009i	$-0.514174\pi$
$751$	−2.43981	−0.0890299	−0.0445150	+	0.999009i	$0.514174\pi$
$752$	−3.00868	−0.109715
$753$	−4.62829	−0.168664
$754$	34.8859	1.27047
$755$	0	0
$756$	0	0
$757$	17.5039	0.636188	0.318094	−	0.948059i	$-0.396957\pi$
$757$	17.5039	0.636188	0.318094	+	0.948059i	$0.396957\pi$
$758$	−9.82440	−0.356838
$759$	19.5557	0.709828
$760$	0	0
$761$	6.06327	0.219793	0.109897	−	0.993943i	$-0.464948\pi$
$761$	6.06327	0.219793	0.109897	+	0.993943i	$0.464948\pi$
$762$	18.4434	0.668134
$763$	0	0
$764$	−19.7269	−0.713693
$765$	0	0
$766$	−30.6647	−1.10796
$767$	49.6551	1.79294
$768$	−1.00000	−0.0360844
$769$	6.94628	0.250489	0.125245	−	0.992126i	$-0.460028\pi$
$769$	6.94628	0.250489	0.125245	+	0.992126i	$0.460028\pi$
$770$	0	0
$771$	25.5671	0.920777
$772$	−4.78654	−0.172271
$773$	−14.8597	−0.534466	−0.267233	−	0.963632i	$-0.586109\pi$
$773$	−14.8597	−0.534466	−0.267233	+	0.963632i	$0.586109\pi$
$774$	4.90755	0.176398
$775$	0	0
$776$	−12.0624	−0.433015
$777$	0	0
$778$	23.4476	0.840637
$779$	13.3146	0.477045
$780$	0	0
$781$	−16.6101	−0.594355
$782$	−16.1002	−0.575744
$783$	9.16246	0.327440
$784$	0	0
$785$	0	0
$786$	0.615405	0.0219508
$787$	−16.7683	−0.597726	−0.298863	−	0.954296i	$-0.596607\pi$
$787$	−16.7683	−0.597726	−0.298863	+	0.954296i	$0.596607\pi$
$788$	17.1997	0.612714
$789$	−20.5972	−0.733279
$790$	0	0
$791$	0	0
$792$	−2.39327	−0.0850411
$793$	49.8436	1.77000
$794$	35.3761	1.25545
$795$	0	0
$796$	−15.5056	−0.549582
$797$	37.5481	1.33002	0.665011	−	0.746833i	$-0.268427\pi$
$797$	37.5481	1.33002	0.665011	+	0.746833i	$0.268427\pi$
$798$	0	0
$799$	5.92824	0.209726
$800$	0	0
$801$	−1.97949	−0.0699420
$802$	−9.04145	−0.319265
$803$	−1.21258	−0.0427911
$804$	5.77786	0.203770
$805$	0	0
$806$	6.11760	0.215483
$807$	−2.23799	−0.0787809
$808$	−7.86672	−0.276750
$809$	12.5014	0.439526	0.219763	−	0.975553i	$-0.429472\pi$
$809$	12.5014	0.439526	0.219763	+	0.975553i	$0.429472\pi$
$810$	0	0
$811$	30.4470	1.06914	0.534569	−	0.845125i	$-0.320474\pi$
$811$	30.4470	1.06914	0.534569	+	0.845125i	$0.320474\pi$
$812$	0	0
$813$	−19.1616	−0.672026
$814$	−3.02410	−0.105995
$815$	0	0
$816$	1.97038	0.0689771
$817$	20.4700	0.716156
$818$	15.4379	0.539773
$819$	0	0
$820$	0	0
$821$	38.2775	1.33589	0.667947	−	0.744209i	$-0.267173\pi$
$821$	38.2775	1.33589	0.667947	+	0.744209i	$0.267173\pi$
$822$	7.82799	0.273032
$823$	23.6213	0.823386	0.411693	−	0.911323i	$-0.364938\pi$
$823$	23.6213	0.823386	0.411693	+	0.911323i	$0.364938\pi$
$824$	−5.89887	−0.205497
$825$	0	0
$826$	0	0
$827$	−34.5652	−1.20195	−0.600975	−	0.799268i	$-0.705221\pi$
$827$	−34.5652	−1.20195	−0.600975	+	0.799268i	$0.705221\pi$
$828$	8.17113	0.283967
$829$	27.2086	0.944992	0.472496	−	0.881333i	$-0.343353\pi$
$829$	27.2086	0.944992	0.472496	+	0.881333i	$0.343353\pi$
$830$	0	0
$831$	−3.67674	−0.127545
$832$	−3.80748	−0.132001
$833$	0	0
$834$	−6.15378	−0.213088
$835$	0	0
$836$	−9.98265	−0.345257
$837$	1.60673	0.0555367
$838$	26.6274	0.919829
$839$	1.49861	0.0517377	0.0258689	−	0.999665i	$-0.491765\pi$
$839$	1.49861	0.0517377	0.0258689	+	0.999665i	$0.491765\pi$
$840$	0	0
$841$	54.9507	1.89485
$842$	5.04591	0.173894
$843$	29.8280	1.02733
$844$	−23.8980	−0.822603
$845$	0	0
$846$	−3.00868	−0.103440
$847$	0	0
$848$	12.1711	0.417958
$849$	−4.28961	−0.147219
$850$	0	0
$851$	10.3249	0.353934
$852$	−6.94032	−0.237772
$853$	38.9306	1.33296	0.666479	−	0.745524i	$-0.267801\pi$
$853$	38.9306	1.33296	0.666479	+	0.745524i	$0.267801\pi$
$854$	0	0
$855$	0	0
$856$	3.75798	0.128445
$857$	18.5987	0.635319	0.317659	−	0.948205i	$-0.397103\pi$
$857$	18.5987	0.635319	0.317659	+	0.948205i	$0.397103\pi$
$858$	−9.11233	−0.311090
$859$	9.09945	0.310469	0.155235	−	0.987878i	$-0.450387\pi$
$859$	9.09945	0.310469	0.155235	+	0.987878i	$0.450387\pi$
$860$	0	0
$861$	0	0
$862$	−25.2835	−0.861158
$863$	10.4616	0.356118	0.178059	−	0.984020i	$-0.443018\pi$
$863$	10.4616	0.356118	0.178059	+	0.984020i	$0.443018\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−27.0063	−0.917711
$867$	13.1176	0.445497
$868$	0	0
$869$	29.7389	1.00882
$870$	0	0
$871$	21.9991	0.745412
$872$	7.14257	0.241878
$873$	−12.0624	−0.408250
$874$	34.0829	1.15287
$875$	0	0
$876$	−0.506664	−0.0171186
$877$	−5.29829	−0.178910	−0.0894552	−	0.995991i	$-0.528513\pi$
$877$	−5.29829	−0.178910	−0.0894552	+	0.995991i	$0.528513\pi$
$878$	−27.9316	−0.942647
$879$	−24.5940	−0.829536
$880$	0	0
$881$	2.68857	0.0905802	0.0452901	−	0.998974i	$-0.485579\pi$
$881$	2.68857	0.0905802	0.0452901	+	0.998974i	$0.485579\pi$
$882$	0	0
$883$	−26.4277	−0.889363	−0.444681	−	0.895689i	$-0.646683\pi$
$883$	−26.4277	−0.889363	−0.444681	+	0.895689i	$0.646683\pi$
$884$	7.50219	0.252326
$885$	0	0
$886$	10.7692	0.361798
$887$	−8.18149	−0.274708	−0.137354	−	0.990522i	$-0.543860\pi$
$887$	−8.18149	−0.274708	−0.137354	+	0.990522i	$0.543860\pi$
$888$	−1.26358	−0.0424031
$889$	0	0
$890$	0	0
$891$	−2.39327	−0.0801776
$892$	−0.870315	−0.0291403
$893$	−12.5496	−0.419956
$894$	7.53584	0.252036
$895$	0	0
$896$	0	0
$897$	31.1115	1.03878
$898$	−15.1288	−0.504855
$899$	14.7216	0.490993
$900$	0	0
$901$	−23.9818	−0.798949
$902$	−7.63950	−0.254368
$903$	0	0
$904$	−17.3837	−0.578174
$905$	0	0
$906$	−9.04145	−0.300382
$907$	−3.80028	−0.126186	−0.0630932	−	0.998008i	$-0.520097\pi$
$907$	−3.80028	−0.126186	−0.0630932	+	0.998008i	$0.520097\pi$
$908$	−14.6854	−0.487353
$909$	−7.86672	−0.260923
$910$	0	0
$911$	39.1590	1.29740	0.648699	−	0.761045i	$-0.275314\pi$
$911$	39.1590	1.29740	0.648699	+	0.761045i	$0.275314\pi$
$912$	−4.17113	−0.138120
$913$	−4.54452	−0.150402
$914$	21.4711	0.710200
$915$	0	0
$916$	−24.0379	−0.794233
$917$	0	0
$918$	1.97038	0.0650323
$919$	−13.8462	−0.456745	−0.228372	−	0.973574i	$-0.573340\pi$
$919$	−13.8462	−0.456745	−0.228372	+	0.973574i	$0.573340\pi$
$920$	0	0
$921$	−2.14089	−0.0705449
$922$	30.4465	1.00270
$923$	−26.4252	−0.869795
$924$	0	0
$925$	0	0
$926$	−40.2005	−1.32107
$927$	−5.89887	−0.193744
$928$	−9.16246	−0.300772
$929$	13.8030	0.452862	0.226431	−	0.974027i	$-0.427294\pi$
$929$	13.8030	0.452862	0.226431	+	0.974027i	$0.427294\pi$
$930$	0	0
$931$	0	0
$932$	10.5445	0.345397
$933$	13.9991	0.458311
$934$	−7.84175	−0.256590
$935$	0	0
$936$	−3.80748	−0.124451
$937$	29.1062	0.950858	0.475429	−	0.879754i	$-0.342293\pi$
$937$	29.1062	0.950858	0.475429	+	0.879754i	$0.342293\pi$
$938$	0	0
$939$	−33.0727	−1.07929
$940$	0	0
$941$	26.9934	0.879960	0.439980	−	0.898008i	$-0.354986\pi$
$941$	26.9934	0.879960	0.439980	+	0.898008i	$0.354986\pi$
$942$	−22.6640	−0.738434
$943$	26.0829	0.849376
$944$	−13.0414	−0.424463
$945$	0	0
$946$	−11.7451	−0.381866
$947$	32.5005	1.05612	0.528062	−	0.849206i	$-0.322919\pi$
$947$	32.5005	1.05612	0.528062	+	0.849206i	$0.322919\pi$
$948$	12.4260	0.403579
$949$	−1.92911	−0.0626217
$950$	0	0
$951$	−7.38459	−0.239462
$952$	0	0
$953$	41.4364	1.34226	0.671128	−	0.741342i	$-0.265810\pi$
$953$	41.4364	1.34226	0.671128	+	0.741342i	$0.265810\pi$
$954$	12.1711	0.394055
$955$	0	0
$956$	−4.01289	−0.129786
$957$	−21.9282	−0.708839
$958$	−27.7745	−0.897352
$959$	0	0
$960$	0	0
$961$	−28.4184	−0.916723
$962$	−4.81108	−0.155115
$963$	3.75798	0.121099
$964$	−0.543899	−0.0175178
$965$	0	0
$966$	0	0
$967$	−23.0294	−0.740577	−0.370288	−	0.928917i	$-0.620741\pi$
$967$	−23.0294	−0.740577	−0.370288	+	0.928917i	$0.620741\pi$
$968$	−5.27226	−0.169457
$969$	8.21872	0.264023
$970$	0	0
$971$	5.06326	0.162488	0.0812439	−	0.996694i	$-0.474111\pi$
$971$	5.06326	0.162488	0.0812439	+	0.996694i	$0.474111\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	12.5263	0.401368
$975$	0	0
$976$	−13.0910	−0.419031
$977$	−29.6213	−0.947669	−0.473834	−	0.880614i	$-0.657130\pi$
$977$	−29.6213	−0.947669	−0.473834	+	0.880614i	$0.657130\pi$
$978$	−10.2039	−0.326285
$979$	4.73747	0.151410
$980$	0	0
$981$	7.14257	0.228045
$982$	−24.3924	−0.778393
$983$	8.18149	0.260949	0.130474	−	0.991452i	$-0.458350\pi$
$983$	8.18149	0.260949	0.130474	+	0.991452i	$0.458350\pi$
$984$	−3.19208	−0.101760
$985$	0	0
$986$	18.0535	0.574942
$987$	0	0
$988$	−15.8815	−0.505259
$989$	40.1002	1.27511
$990$	0	0
$991$	16.7554	0.532254	0.266127	−	0.963938i	$-0.414256\pi$
$991$	16.7554	0.532254	0.266127	+	0.963938i	$0.414256\pi$
$992$	−1.60673	−0.0510137
$993$	−21.5549	−0.684023
$994$	0	0
$995$	0	0
$996$	−1.89887	−0.0601681
$997$	4.62390	0.146440	0.0732202	−	0.997316i	$-0.476672\pi$
$997$	4.62390	0.146440	0.0732202	+	0.997316i	$0.476672\pi$
$998$	−23.6560	−0.748817
$999$	−1.26358	−0.0399780

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.dt.1.2		4
5.2	odd	4		1470.2.g.j.589.5	yes	8
5.3	odd	4		1470.2.g.j.589.1	✓	8
5.4	even	2		7350.2.a.ds.1.2		4
7.6	odd	2		7350.2.a.du.1.2		4
35.2	odd	12		1470.2.n.l.949.7		16
35.3	even	12		1470.2.n.k.79.6		16
35.12	even	12		1470.2.n.k.949.6		16
35.13	even	4		1470.2.g.k.589.4	yes	8
35.17	even	12		1470.2.n.k.79.2		16
35.18	odd	12		1470.2.n.l.79.7		16
35.23	odd	12		1470.2.n.l.949.3		16
35.27	even	4		1470.2.g.k.589.8	yes	8
35.32	odd	12		1470.2.n.l.79.3		16
35.33	even	12		1470.2.n.k.949.2		16
35.34	odd	2		7350.2.a.dr.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1470.2.g.j.589.1	✓	8	5.3	odd	4
1470.2.g.j.589.5	yes	8	5.2	odd	4
1470.2.g.k.589.4	yes	8	35.13	even	4
1470.2.g.k.589.8	yes	8	35.27	even	4
1470.2.n.k.79.2		16	35.17	even	12
1470.2.n.k.79.6		16	35.3	even	12
1470.2.n.k.949.2		16	35.33	even	12
1470.2.n.k.949.6		16	35.12	even	12
1470.2.n.l.79.3		16	35.32	odd	12
1470.2.n.l.79.7		16	35.18	odd	12
1470.2.n.l.949.3		16	35.23	odd	12
1470.2.n.l.949.7		16	35.2	odd	12
7350.2.a.dr.1.2		4	35.34	odd	2
7350.2.a.ds.1.2		4	5.4	even	2
7350.2.a.dt.1.2		4	1.1	even	1	trivial
7350.2.a.du.1.2		4	7.6	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:	\( N \)	\(=\)	\( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7350.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.39327 q^{11} -1.00000 q^{12} -3.80748 q^{13} +1.00000 q^{16} -1.97038 q^{17} +1.00000 q^{18} +4.17113 q^{19} -2.39327 q^{22} +8.17113 q^{23} -1.00000 q^{24} -3.80748 q^{26} -1.00000 q^{27} -9.16246 q^{29} -1.60673 q^{31} +1.00000 q^{32} +2.39327 q^{33} -1.97038 q^{34} +1.00000 q^{36} +1.26358 q^{37} +4.17113 q^{38} +3.80748 q^{39} +3.19208 q^{41} +4.90755 q^{43} -2.39327 q^{44} +8.17113 q^{46} -3.00868 q^{47} -1.00000 q^{48} +1.97038 q^{51} -3.80748 q^{52} +12.1711 q^{53} -1.00000 q^{54} -4.17113 q^{57} -9.16246 q^{58} -13.0414 q^{59} -13.0910 q^{61} -1.60673 q^{62} +1.00000 q^{64} +2.39327 q^{66} -5.77786 q^{67} -1.97038 q^{68} -8.17113 q^{69} +6.94032 q^{71} +1.00000 q^{72} +0.506664 q^{73} +1.26358 q^{74} +4.17113 q^{76} +3.80748 q^{78} -12.4260 q^{79} +1.00000 q^{81} +3.19208 q^{82} +1.89887 q^{83} +4.90755 q^{86} +9.16246 q^{87} -2.39327 q^{88} -1.97949 q^{89} +8.17113 q^{92} +1.60673 q^{93} -3.00868 q^{94} -1.00000 q^{96} -12.0624 q^{97} -2.39327 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^6 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} + 4 q^{8} + 4 q^{9} - 4 q^{12} + 4 q^{16} - 4 q^{17} + 4 q^{18} - 12 q^{19} + 4 q^{23} - 4 q^{24} - 4 q^{27} - 8 q^{29} - 16 q^{31} + 4 q^{32} - 4 q^{34} + 4 q^{36} - 8 q^{37} - 12 q^{38} - 12 q^{41} + 4 q^{43} + 4 q^{46} - 12 q^{47} - 4 q^{48} + 4 q^{51} + 20 q^{53} - 4 q^{54} + 12 q^{57} - 8 q^{58} - 20 q^{59} - 12 q^{61} - 16 q^{62} + 4 q^{64} - 4 q^{67} - 4 q^{68} - 4 q^{69} - 20 q^{71} + 4 q^{72} + 12 q^{73} - 8 q^{74} - 12 q^{76} - 8 q^{79} + 4 q^{81} - 12 q^{82} - 8 q^{83} + 4 q^{86} + 8 q^{87} - 44 q^{89} + 4 q^{92} + 16 q^{93} - 12 q^{94} - 4 q^{96} - 20 q^{97}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^6 + 4 * q^8 + 4 * q^9 - 4 * q^12 + 4 * q^16 - 4 * q^17 + 4 * q^18 - 12 * q^19 + 4 * q^23 - 4 * q^24 - 4 * q^27 - 8 * q^29 - 16 * q^31 + 4 * q^32 - 4 * q^34 + 4 * q^36 - 8 * q^37 - 12 * q^38 - 12 * q^41 + 4 * q^43 + 4 * q^46 - 12 * q^47 - 4 * q^48 + 4 * q^51 + 20 * q^53 - 4 * q^54 + 12 * q^57 - 8 * q^58 - 20 * q^59 - 12 * q^61 - 16 * q^62 + 4 * q^64 - 4 * q^67 - 4 * q^68 - 4 * q^69 - 20 * q^71 + 4 * q^72 + 12 * q^73 - 8 * q^74 - 12 * q^76 - 8 * q^79 + 4 * q^81 - 12 * q^82 - 8 * q^83 + 4 * q^86 + 8 * q^87 - 44 * q^89 + 4 * q^92 + 16 * q^93 - 12 * q^94 - 4 * q^96 - 20 * q^97

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