LMFDB - Embedded newform 8752.2.a.x.1.29

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8752,2,Mod(1,8752)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8752, 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("8752.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8752 = 2^{4} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8752.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:	$69.8850718490$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	no (minimal twist has level 4376)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.29
Character	$\chi$	$=$	8752.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.24040 q^{3} -2.27349 q^{5} -0.587186 q^{7} +7.50019 q^{9} +O(q^{10})$	$q+3.24040 q^{3} -2.27349 q^{5} -0.587186 q^{7} +7.50019 q^{9} +0.562791 q^{11} +1.76770 q^{13} -7.36701 q^{15} -3.97692 q^{17} +5.97060 q^{19} -1.90272 q^{21} -0.0297800 q^{23} +0.168753 q^{25} +14.5824 q^{27} +2.02198 q^{29} -2.64436 q^{31} +1.82367 q^{33} +1.33496 q^{35} +6.81586 q^{37} +5.72804 q^{39} +8.90757 q^{41} +5.63983 q^{43} -17.0516 q^{45} -8.87979 q^{47} -6.65521 q^{49} -12.8868 q^{51} +11.8378 q^{53} -1.27950 q^{55} +19.3471 q^{57} +11.2148 q^{59} -7.50492 q^{61} -4.40400 q^{63} -4.01884 q^{65} -3.77078 q^{67} -0.0964993 q^{69} -7.23903 q^{71} +4.75020 q^{73} +0.546828 q^{75} -0.330463 q^{77} -14.7615 q^{79} +24.7523 q^{81} +6.66318 q^{83} +9.04149 q^{85} +6.55202 q^{87} -4.60576 q^{89} -1.03797 q^{91} -8.56877 q^{93} -13.5741 q^{95} +0.593574 q^{97} +4.22104 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 8 q^{3} - 11 q^{5} + 10 q^{7} + 39 q^{9}+O(q^{10}) $ 31 * q + 8 * q^3 - 11 * q^5 + 10 * q^7 + 39 * q^9	$ 31 q + 8 q^{3} - 11 q^{5} + 10 q^{7} + 39 q^{9} + 11 q^{11} - 20 q^{13} + 11 q^{15} + 26 q^{17} + 32 q^{19} - 23 q^{21} + 29 q^{23} + 26 q^{25} + 53 q^{27} - 6 q^{29} + 53 q^{31} + 8 q^{33} + 13 q^{35} - 26 q^{37} + 24 q^{39} - 6 q^{41} + 11 q^{43} - 12 q^{45} + 41 q^{47} + 29 q^{49} + 35 q^{51} - 17 q^{53} + 10 q^{55} + 9 q^{57} + 27 q^{59} - 62 q^{61} + 34 q^{63} + 4 q^{65} + 28 q^{67} - 20 q^{69} + 39 q^{71} + 16 q^{73} + 36 q^{75} - 19 q^{77} + 39 q^{79} + 35 q^{81} + 32 q^{83} - 58 q^{85} + 38 q^{87} + 61 q^{89} + 25 q^{91} - q^{93} + 12 q^{95} + 85 q^{99}+O(q^{100}) $ 31 * q + 8 * q^3 - 11 * q^5 + 10 * q^7 + 39 * q^9 + 11 * q^11 - 20 * q^13 + 11 * q^15 + 26 * q^17 + 32 * q^19 - 23 * q^21 + 29 * q^23 + 26 * q^25 + 53 * q^27 - 6 * q^29 + 53 * q^31 + 8 * q^33 + 13 * q^35 - 26 * q^37 + 24 * q^39 - 6 * q^41 + 11 * q^43 - 12 * q^45 + 41 * q^47 + 29 * q^49 + 35 * q^51 - 17 * q^53 + 10 * q^55 + 9 * q^57 + 27 * q^59 - 62 * q^61 + 34 * q^63 + 4 * q^65 + 28 * q^67 - 20 * q^69 + 39 * q^71 + 16 * q^73 + 36 * q^75 - 19 * q^77 + 39 * q^79 + 35 * q^81 + 32 * q^83 - 58 * q^85 + 38 * q^87 + 61 * q^89 + 25 * q^91 - q^93 + 12 * q^95 + 85 * 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
$3$	3.24040	1.87085	0.935423	−	0.353531i	$-0.115019\pi$
$3$	3.24040	1.87085	0.935423	+	0.353531i	$0.115019\pi$
$4$	0	0
$5$	−2.27349	−1.01674	−0.508368	−	0.861140i	$-0.669751\pi$
$5$	−2.27349	−1.01674	−0.508368	+	0.861140i	$0.669751\pi$
$6$	0	0
$7$	−0.587186	−0.221935	−0.110968	−	0.993824i	$-0.535395\pi$
$7$	−0.587186	−0.221935	−0.110968	+	0.993824i	$0.535395\pi$
$8$	0	0
$9$	7.50019	2.50006
$10$	0	0
$11$	0.562791	0.169688	0.0848440	−	0.996394i	$-0.472961\pi$
$11$	0.562791	0.169688	0.0848440	+	0.996394i	$0.472961\pi$
$12$	0	0
$13$	1.76770	0.490271	0.245135	−	0.969489i	$-0.421168\pi$
$13$	1.76770	0.490271	0.245135	+	0.969489i	$0.421168\pi$
$14$	0	0
$15$	−7.36701	−1.90215
$16$	0	0
$17$	−3.97692	−0.964545	−0.482273	−	0.876021i	$-0.660189\pi$
$17$	−3.97692	−0.964545	−0.482273	+	0.876021i	$0.660189\pi$
$18$	0	0
$19$	5.97060	1.36975	0.684875	−	0.728660i	$-0.259857\pi$
$19$	5.97060	1.36975	0.684875	+	0.728660i	$0.259857\pi$
$20$	0	0
$21$	−1.90272	−0.415207
$22$	0	0
$23$	−0.0297800	−0.00620957	−0.00310478	−	0.999995i	$-0.500988\pi$
$23$	−0.0297800	−0.00620957	−0.00310478	+	0.999995i	$0.500988\pi$
$24$	0	0
$25$	0.168753	0.0337506
$26$	0	0
$27$	14.5824	2.80639
$28$	0	0
$29$	2.02198	0.375472	0.187736	−	0.982220i	$-0.439885\pi$
$29$	2.02198	0.375472	0.187736	+	0.982220i	$0.439885\pi$
$30$	0	0
$31$	−2.64436	−0.474941	−0.237470	−	0.971395i	$-0.576318\pi$
$31$	−2.64436	−0.474941	−0.237470	+	0.971395i	$0.576318\pi$
$32$	0	0
$33$	1.82367	0.317460
$34$	0	0
$35$	1.33496	0.225649
$36$	0	0
$37$	6.81586	1.12052	0.560260	−	0.828316i	$-0.310701\pi$
$37$	6.81586	1.12052	0.560260	+	0.828316i	$0.310701\pi$
$38$	0	0
$39$	5.72804	0.917221
$40$	0	0
$41$	8.90757	1.39113	0.695564	−	0.718464i	$-0.255155\pi$
$41$	8.90757	1.39113	0.695564	+	0.718464i	$0.255155\pi$
$42$	0	0
$43$	5.63983	0.860066	0.430033	−	0.902813i	$-0.358502\pi$
$43$	5.63983	0.860066	0.430033	+	0.902813i	$0.358502\pi$
$44$	0	0
$45$	−17.0516	−2.54190
$46$	0	0
$47$	−8.87979	−1.29525	−0.647625	−	0.761959i	$-0.724238\pi$
$47$	−8.87979	−1.29525	−0.647625	+	0.761959i	$0.724238\pi$
$48$	0	0
$49$	−6.65521	−0.950745
$50$	0	0
$51$	−12.8868	−1.80452
$52$	0	0
$53$	11.8378	1.62604	0.813021	−	0.582235i	$-0.197821\pi$
$53$	11.8378	1.62604	0.813021	+	0.582235i	$0.197821\pi$
$54$	0	0
$55$	−1.27950	−0.172528
$56$	0	0
$57$	19.3471	2.56259
$58$	0	0
$59$	11.2148	1.46004	0.730019	−	0.683427i	$-0.239511\pi$
$59$	11.2148	1.46004	0.730019	+	0.683427i	$0.239511\pi$
$60$	0	0
$61$	−7.50492	−0.960906	−0.480453	−	0.877020i	$-0.659528\pi$
$61$	−7.50492	−0.960906	−0.480453	+	0.877020i	$0.659528\pi$
$62$	0	0
$63$	−4.40400	−0.554852
$64$	0	0
$65$	−4.01884	−0.498475
$66$	0	0
$67$	−3.77078	−0.460673	−0.230337	−	0.973111i	$-0.573983\pi$
$67$	−3.77078	−0.460673	−0.230337	+	0.973111i	$0.573983\pi$
$68$	0	0
$69$	−0.0964993	−0.0116171
$70$	0	0
$71$	−7.23903	−0.859115	−0.429557	−	0.903040i	$-0.641330\pi$
$71$	−7.23903	−0.859115	−0.429557	+	0.903040i	$0.641330\pi$
$72$	0	0
$73$	4.75020	0.555969	0.277984	−	0.960586i	$-0.410334\pi$
$73$	4.75020	0.555969	0.277984	+	0.960586i	$0.410334\pi$
$74$	0	0
$75$	0.546828	0.0631422
$76$	0	0
$77$	−0.330463	−0.0376597
$78$	0	0
$79$	−14.7615	−1.66080	−0.830398	−	0.557171i	$-0.811887\pi$
$79$	−14.7615	−1.66080	−0.830398	+	0.557171i	$0.811887\pi$
$80$	0	0
$81$	24.7523	2.75025
$82$	0	0
$83$	6.66318	0.731379	0.365689	−	0.930737i	$-0.380833\pi$
$83$	6.66318	0.731379	0.365689	+	0.930737i	$0.380833\pi$
$84$	0	0
$85$	9.04149	0.980687
$86$	0	0
$87$	6.55202	0.702450
$88$	0	0
$89$	−4.60576	−0.488209	−0.244105	−	0.969749i	$-0.578494\pi$
$89$	−4.60576	−0.488209	−0.244105	+	0.969749i	$0.578494\pi$
$90$	0	0
$91$	−1.03797	−0.108808
$92$	0	0
$93$	−8.56877	−0.888540
$94$	0	0
$95$	−13.5741	−1.39267
$96$	0	0
$97$	0.593574	0.0602683	0.0301341	−	0.999546i	$-0.490407\pi$
$97$	0.593574	0.0602683	0.0301341	+	0.999546i	$0.490407\pi$
$98$	0	0
$99$	4.22104	0.424231
$100$	0	0
$101$	6.41705	0.638520	0.319260	−	0.947667i	$-0.396566\pi$
$101$	6.41705	0.638520	0.319260	+	0.947667i	$0.396566\pi$
$102$	0	0
$103$	−9.73359	−0.959079	−0.479540	−	0.877520i	$-0.659196\pi$
$103$	−9.73359	−0.959079	−0.479540	+	0.877520i	$0.659196\pi$
$104$	0	0
$105$	4.32580	0.422155
$106$	0	0
$107$	−1.13664	−0.109883	−0.0549415	−	0.998490i	$-0.517497\pi$
$107$	−1.13664	−0.109883	−0.0549415	+	0.998490i	$0.517497\pi$
$108$	0	0
$109$	13.3043	1.27432	0.637162	−	0.770730i	$-0.280108\pi$
$109$	13.3043	1.27432	0.637162	+	0.770730i	$0.280108\pi$
$110$	0	0
$111$	22.0861	2.09632
$112$	0	0
$113$	2.93632	0.276226	0.138113	−	0.990416i	$-0.455896\pi$
$113$	2.93632	0.276226	0.138113	+	0.990416i	$0.455896\pi$
$114$	0	0
$115$	0.0677046	0.00631349
$116$	0	0
$117$	13.2581	1.22571
$118$	0	0
$119$	2.33519	0.214067
$120$	0	0
$121$	−10.6833	−0.971206
$122$	0	0
$123$	28.8641	2.60259
$124$	0	0
$125$	10.9838	0.982420
$126$	0	0
$127$	10.1544	0.901062	0.450531	−	0.892761i	$-0.351235\pi$
$127$	10.1544	0.901062	0.450531	+	0.892761i	$0.351235\pi$
$128$	0	0
$129$	18.2753	1.60905
$130$	0	0
$131$	1.08434	0.0947397	0.0473698	−	0.998877i	$-0.484916\pi$
$131$	1.08434	0.0947397	0.0473698	+	0.998877i	$0.484916\pi$
$132$	0	0
$133$	−3.50585	−0.303996
$134$	0	0
$135$	−33.1530	−2.85335
$136$	0	0
$137$	11.8922	1.01602	0.508008	−	0.861352i	$-0.330382\pi$
$137$	11.8922	1.01602	0.508008	+	0.861352i	$0.330382\pi$
$138$	0	0
$139$	19.1230	1.62199	0.810994	−	0.585054i	$-0.198927\pi$
$139$	19.1230	1.62199	0.810994	+	0.585054i	$0.198927\pi$
$140$	0	0
$141$	−28.7741	−2.42321
$142$	0	0
$143$	0.994844	0.0831930
$144$	0	0
$145$	−4.59695	−0.381756
$146$	0	0
$147$	−21.5656	−1.77870
$148$	0	0
$149$	17.9709	1.47223	0.736116	−	0.676855i	$-0.236658\pi$
$149$	17.9709	1.47223	0.736116	+	0.676855i	$0.236658\pi$
$150$	0	0
$151$	2.15574	0.175432	0.0877158	−	0.996146i	$-0.472043\pi$
$151$	2.15574	0.175432	0.0877158	+	0.996146i	$0.472043\pi$
$152$	0	0
$153$	−29.8277	−2.41142
$154$	0	0
$155$	6.01192	0.482889
$156$	0	0
$157$	−4.13075	−0.329670	−0.164835	−	0.986321i	$-0.552709\pi$
$157$	−4.13075	−0.329670	−0.164835	+	0.986321i	$0.552709\pi$
$158$	0	0
$159$	38.3591	3.04207
$160$	0	0
$161$	0.0174864	0.00137812
$162$	0	0
$163$	15.1626	1.18762	0.593812	−	0.804604i	$-0.297622\pi$
$163$	15.1626	1.18762	0.593812	+	0.804604i	$0.297622\pi$
$164$	0	0
$165$	−4.14609	−0.322773
$166$	0	0
$167$	18.4168	1.42513	0.712567	−	0.701604i	$-0.247533\pi$
$167$	18.4168	1.42513	0.712567	+	0.701604i	$0.247533\pi$
$168$	0	0
$169$	−9.87525	−0.759635
$170$	0	0
$171$	44.7807	3.42446
$172$	0	0
$173$	6.16631	0.468816	0.234408	−	0.972138i	$-0.424685\pi$
$173$	6.16631	0.468816	0.234408	+	0.972138i	$0.424685\pi$
$174$	0	0
$175$	−0.0990894	−0.00749046
$176$	0	0
$177$	36.3403	2.73150
$178$	0	0
$179$	6.84166	0.511370	0.255685	−	0.966760i	$-0.417699\pi$
$179$	6.84166	0.511370	0.255685	+	0.966760i	$0.417699\pi$
$180$	0	0
$181$	−4.24699	−0.315676	−0.157838	−	0.987465i	$-0.550452\pi$
$181$	−4.24699	−0.315676	−0.157838	+	0.987465i	$0.550452\pi$
$182$	0	0
$183$	−24.3189	−1.79771
$184$	0	0
$185$	−15.4958	−1.13927
$186$	0	0
$187$	−2.23818	−0.163672
$188$	0	0
$189$	−8.56258	−0.622836
$190$	0	0
$191$	18.0427	1.30553	0.652763	−	0.757562i	$-0.273610\pi$
$191$	18.0427	1.30553	0.652763	+	0.757562i	$0.273610\pi$
$192$	0	0
$193$	3.33626	0.240149	0.120075	−	0.992765i	$-0.461687\pi$
$193$	3.33626	0.240149	0.120075	+	0.992765i	$0.461687\pi$
$194$	0	0
$195$	−13.0226	−0.932571
$196$	0	0
$197$	−16.0137	−1.14093	−0.570465	−	0.821322i	$-0.693237\pi$
$197$	−16.0137	−1.14093	−0.570465	+	0.821322i	$0.693237\pi$
$198$	0	0
$199$	14.7712	1.04710	0.523552	−	0.851993i	$-0.324606\pi$
$199$	14.7712	1.04710	0.523552	+	0.851993i	$0.324606\pi$
$200$	0	0
$201$	−12.2188	−0.861849
$202$	0	0
$203$	−1.18728	−0.0833305
$204$	0	0
$205$	−20.2513	−1.41441
$206$	0	0
$207$	−0.223356	−0.0155243
$208$	0	0
$209$	3.36020	0.232430
$210$	0	0
$211$	9.34263	0.643173	0.321586	−	0.946880i	$-0.395784\pi$
$211$	9.34263	0.643173	0.321586	+	0.946880i	$0.395784\pi$
$212$	0	0
$213$	−23.4574	−1.60727
$214$	0	0
$215$	−12.8221	−0.874460
$216$	0	0
$217$	1.55273	0.105406
$218$	0	0
$219$	15.3925	1.04013
$220$	0	0
$221$	−7.02999	−0.472888
$222$	0	0
$223$	4.97674	0.333267	0.166634	−	0.986019i	$-0.446710\pi$
$223$	4.97674	0.333267	0.166634	+	0.986019i	$0.446710\pi$
$224$	0	0
$225$	1.26568	0.0843787
$226$	0	0
$227$	−13.0354	−0.865193	−0.432596	−	0.901588i	$-0.642402\pi$
$227$	−13.0354	−0.865193	−0.432596	+	0.901588i	$0.642402\pi$
$228$	0	0
$229$	−15.8286	−1.04598	−0.522992	−	0.852337i	$-0.675184\pi$
$229$	−15.8286	−1.04598	−0.522992	+	0.852337i	$0.675184\pi$
$230$	0	0
$231$	−1.07083	−0.0704555
$232$	0	0
$233$	11.7920	0.772521	0.386261	−	0.922390i	$-0.373767\pi$
$233$	11.7920	0.772521	0.386261	+	0.922390i	$0.373767\pi$
$234$	0	0
$235$	20.1881	1.31693
$236$	0	0
$237$	−47.8331	−3.10709
$238$	0	0
$239$	2.94791	0.190684	0.0953421	−	0.995445i	$-0.469606\pi$
$239$	2.94791	0.190684	0.0953421	+	0.995445i	$0.469606\pi$
$240$	0	0
$241$	−11.6939	−0.753267	−0.376634	−	0.926362i	$-0.622918\pi$
$241$	−11.6939	−0.753267	−0.376634	+	0.926362i	$0.622918\pi$
$242$	0	0
$243$	36.4601	2.33891
$244$	0	0
$245$	15.1306	0.966656
$246$	0	0
$247$	10.5542	0.671548
$248$	0	0
$249$	21.5914	1.36830
$250$	0	0
$251$	19.8117	1.25050	0.625252	−	0.780423i	$-0.284996\pi$
$251$	19.8117	1.25050	0.625252	+	0.780423i	$0.284996\pi$
$252$	0	0
$253$	−0.0167599	−0.00105369
$254$	0	0
$255$	29.2980	1.83471
$256$	0	0
$257$	17.0221	1.06181	0.530905	−	0.847431i	$-0.321852\pi$
$257$	17.0221	1.06181	0.530905	+	0.847431i	$0.321852\pi$
$258$	0	0
$259$	−4.00218	−0.248683
$260$	0	0
$261$	15.1652	0.938704
$262$	0	0
$263$	−3.72903	−0.229942	−0.114971	−	0.993369i	$-0.536677\pi$
$263$	−3.72903	−0.229942	−0.114971	+	0.993369i	$0.536677\pi$
$264$	0	0
$265$	−26.9130	−1.65325
$266$	0	0
$267$	−14.9245	−0.913364
$268$	0	0
$269$	8.78605	0.535695	0.267847	−	0.963461i	$-0.413688\pi$
$269$	8.78605	0.535695	0.267847	+	0.963461i	$0.413688\pi$
$270$	0	0
$271$	−6.66712	−0.404999	−0.202499	−	0.979282i	$-0.564906\pi$
$271$	−6.66712	−0.404999	−0.202499	+	0.979282i	$0.564906\pi$
$272$	0	0
$273$	−3.36342	−0.203564
$274$	0	0
$275$	0.0949728	0.00572707
$276$	0	0
$277$	−32.2600	−1.93831	−0.969157	−	0.246444i	$-0.920738\pi$
$277$	−32.2600	−1.93831	−0.969157	+	0.246444i	$0.920738\pi$
$278$	0	0
$279$	−19.8332	−1.18738
$280$	0	0
$281$	11.8821	0.708826	0.354413	−	0.935089i	$-0.384681\pi$
$281$	11.8821	0.708826	0.354413	+	0.935089i	$0.384681\pi$
$282$	0	0
$283$	−22.8141	−1.35616	−0.678078	−	0.734990i	$-0.737187\pi$
$283$	−22.8141	−1.35616	−0.678078	+	0.734990i	$0.737187\pi$
$284$	0	0
$285$	−43.9855	−2.60548
$286$	0	0
$287$	−5.23039	−0.308740
$288$	0	0
$289$	−1.18410	−0.0696527
$290$	0	0
$291$	1.92342	0.112753
$292$	0	0
$293$	−1.20021	−0.0701173	−0.0350586	−	0.999385i	$-0.511162\pi$
$293$	−1.20021	−0.0701173	−0.0350586	+	0.999385i	$0.511162\pi$
$294$	0	0
$295$	−25.4966	−1.48447
$296$	0	0
$297$	8.20685	0.476210
$298$	0	0
$299$	−0.0526421	−0.00304437
$300$	0	0
$301$	−3.31163	−0.190879
$302$	0	0
$303$	20.7938	1.19457
$304$	0	0
$305$	17.0624	0.976987
$306$	0	0
$307$	−12.7967	−0.730347	−0.365174	−	0.930939i	$-0.618990\pi$
$307$	−12.7967	−0.730347	−0.365174	+	0.930939i	$0.618990\pi$
$308$	0	0
$309$	−31.5407	−1.79429
$310$	0	0
$311$	−17.3789	−0.985467	−0.492733	−	0.870180i	$-0.664002\pi$
$311$	−17.3789	−0.985467	−0.492733	+	0.870180i	$0.664002\pi$
$312$	0	0
$313$	−5.91950	−0.334590	−0.167295	−	0.985907i	$-0.553503\pi$
$313$	−5.91950	−0.334590	−0.167295	+	0.985907i	$0.553503\pi$
$314$	0	0
$315$	10.0125	0.564138
$316$	0	0
$317$	−16.7646	−0.941592	−0.470796	−	0.882242i	$-0.656033\pi$
$317$	−16.7646	−0.941592	−0.470796	+	0.882242i	$0.656033\pi$
$318$	0	0
$319$	1.13795	0.0637131
$320$	0	0
$321$	−3.68316	−0.205574
$322$	0	0
$323$	−23.7446	−1.32119
$324$	0	0
$325$	0.298304	0.0165469
$326$	0	0
$327$	43.1114	2.38406
$328$	0	0
$329$	5.21408	0.287462
$330$	0	0
$331$	−28.8384	−1.58510	−0.792550	−	0.609807i	$-0.791247\pi$
$331$	−28.8384	−1.58510	−0.792550	+	0.609807i	$0.791247\pi$
$332$	0	0
$333$	51.1203	2.80137
$334$	0	0
$335$	8.57282	0.468383
$336$	0	0
$337$	−23.7748	−1.29509	−0.647547	−	0.762025i	$-0.724205\pi$
$337$	−23.7748	−1.29509	−0.647547	+	0.762025i	$0.724205\pi$
$338$	0	0
$339$	9.51486	0.516776
$340$	0	0
$341$	−1.48822	−0.0805917
$342$	0	0
$343$	8.01814	0.432939
$344$	0	0
$345$	0.219390	0.0118116
$346$	0	0
$347$	17.1041	0.918197	0.459099	−	0.888385i	$-0.348172\pi$
$347$	17.1041	0.918197	0.459099	+	0.888385i	$0.348172\pi$
$348$	0	0
$349$	−9.57720	−0.512656	−0.256328	−	0.966590i	$-0.582513\pi$
$349$	−9.57720	−0.512656	−0.256328	+	0.966590i	$0.582513\pi$
$350$	0	0
$351$	25.7773	1.37589
$352$	0	0
$353$	5.28253	0.281161	0.140580	−	0.990069i	$-0.455103\pi$
$353$	5.28253	0.281161	0.140580	+	0.990069i	$0.455103\pi$
$354$	0	0
$355$	16.4579	0.873492
$356$	0	0
$357$	7.56695	0.400486
$358$	0	0
$359$	7.10365	0.374916	0.187458	−	0.982273i	$-0.439975\pi$
$359$	7.10365	0.374916	0.187458	+	0.982273i	$0.439975\pi$
$360$	0	0
$361$	16.6481	0.876216
$362$	0	0
$363$	−34.6181	−1.81698
$364$	0	0
$365$	−10.7995	−0.565273
$366$	0	0
$367$	23.4431	1.22372	0.611859	−	0.790967i	$-0.290422\pi$
$367$	23.4431	1.22372	0.611859	+	0.790967i	$0.290422\pi$
$368$	0	0
$369$	66.8084	3.47791
$370$	0	0
$371$	−6.95096	−0.360876
$372$	0	0
$373$	25.6748	1.32939	0.664696	−	0.747114i	$-0.268561\pi$
$373$	25.6748	1.32939	0.664696	+	0.747114i	$0.268561\pi$
$374$	0	0
$375$	35.5919	1.83796
$376$	0	0
$377$	3.57424	0.184083
$378$	0	0
$379$	24.4131	1.25402	0.627009	−	0.779012i	$-0.284279\pi$
$379$	24.4131	1.25402	0.627009	+	0.779012i	$0.284279\pi$
$380$	0	0
$381$	32.9045	1.68575
$382$	0	0
$383$	−21.7677	−1.11228	−0.556139	−	0.831090i	$-0.687718\pi$
$383$	−21.7677	−1.11228	−0.556139	+	0.831090i	$0.687718\pi$
$384$	0	0
$385$	0.751304	0.0382900
$386$	0	0
$387$	42.2998	2.15022
$388$	0	0
$389$	12.9005	0.654080	0.327040	−	0.945011i	$-0.393949\pi$
$389$	12.9005	0.654080	0.327040	+	0.945011i	$0.393949\pi$
$390$	0	0
$391$	0.118433	0.00598941
$392$	0	0
$393$	3.51371	0.177243
$394$	0	0
$395$	33.5601	1.68859
$396$	0	0
$397$	−8.75062	−0.439181	−0.219591	−	0.975592i	$-0.570472\pi$
$397$	−8.75062	−0.439181	−0.219591	+	0.975592i	$0.570472\pi$
$398$	0	0
$399$	−11.3604	−0.568729
$400$	0	0
$401$	−31.1301	−1.55456	−0.777281	−	0.629154i	$-0.783401\pi$
$401$	−31.1301	−1.55456	−0.777281	+	0.629154i	$0.783401\pi$
$402$	0	0
$403$	−4.67442	−0.232849
$404$	0	0
$405$	−56.2741	−2.79628
$406$	0	0
$407$	3.83591	0.190139
$408$	0	0
$409$	22.9962	1.13709	0.568545	−	0.822652i	$-0.307506\pi$
$409$	22.9962	1.13709	0.568545	+	0.822652i	$0.307506\pi$
$410$	0	0
$411$	38.5354	1.90081
$412$	0	0
$413$	−6.58514	−0.324034
$414$	0	0
$415$	−15.1487	−0.743619
$416$	0	0
$417$	61.9661	3.03449
$418$	0	0
$419$	37.0817	1.81156	0.905781	−	0.423747i	$-0.139285\pi$
$419$	37.0817	1.81156	0.905781	+	0.423747i	$0.139285\pi$
$420$	0	0
$421$	−31.8735	−1.55342	−0.776710	−	0.629858i	$-0.783113\pi$
$421$	−31.8735	−1.55342	−0.776710	+	0.629858i	$0.783113\pi$
$422$	0	0
$423$	−66.6001	−3.23821
$424$	0	0
$425$	−0.671118	−0.0325540
$426$	0	0
$427$	4.40678	0.213259
$428$	0	0
$429$	3.22369	0.155641
$430$	0	0
$431$	30.5841	1.47318	0.736592	−	0.676337i	$-0.236434\pi$
$431$	30.5841	1.47318	0.736592	+	0.676337i	$0.236434\pi$
$432$	0	0
$433$	−21.8714	−1.05107	−0.525537	−	0.850771i	$-0.676136\pi$
$433$	−21.8714	−1.05107	−0.525537	+	0.850771i	$0.676136\pi$
$434$	0	0
$435$	−14.8959	−0.714206
$436$	0	0
$437$	−0.177805	−0.00850556
$438$	0	0
$439$	−5.10090	−0.243452	−0.121726	−	0.992564i	$-0.538843\pi$
$439$	−5.10090	−0.243452	−0.121726	+	0.992564i	$0.538843\pi$
$440$	0	0
$441$	−49.9154	−2.37692
$442$	0	0
$443$	−14.0318	−0.666671	−0.333336	−	0.942808i	$-0.608174\pi$
$443$	−14.0318	−0.666671	−0.333336	+	0.942808i	$0.608174\pi$
$444$	0	0
$445$	10.4711	0.496379
$446$	0	0
$447$	58.2329	2.75432
$448$	0	0
$449$	−35.6533	−1.68258	−0.841291	−	0.540583i	$-0.818204\pi$
$449$	−35.6533	−1.68258	−0.841291	+	0.540583i	$0.818204\pi$
$450$	0	0
$451$	5.01310	0.236058
$452$	0	0
$453$	6.98546	0.328205
$454$	0	0
$455$	2.35980	0.110629
$456$	0	0
$457$	34.9710	1.63588	0.817938	−	0.575306i	$-0.195117\pi$
$457$	34.9710	1.63588	0.817938	+	0.575306i	$0.195117\pi$
$458$	0	0
$459$	−57.9931	−2.70689
$460$	0	0
$461$	−12.0084	−0.559286	−0.279643	−	0.960104i	$-0.590216\pi$
$461$	−12.0084	−0.559286	−0.279643	+	0.960104i	$0.590216\pi$
$462$	0	0
$463$	−15.5326	−0.721862	−0.360931	−	0.932593i	$-0.617541\pi$
$463$	−15.5326	−0.721862	−0.360931	+	0.932593i	$0.617541\pi$
$464$	0	0
$465$	19.4810	0.903410
$466$	0	0
$467$	1.36267	0.0630569	0.0315284	−	0.999503i	$-0.489963\pi$
$467$	1.36267	0.0630569	0.0315284	+	0.999503i	$0.489963\pi$
$468$	0	0
$469$	2.21414	0.102240
$470$	0	0
$471$	−13.3853	−0.616761
$472$	0	0
$473$	3.17405	0.145943
$474$	0	0
$475$	1.00756	0.0462299
$476$	0	0
$477$	88.7855	4.06521
$478$	0	0
$479$	6.33484	0.289446	0.144723	−	0.989472i	$-0.453771\pi$
$479$	6.33484	0.289446	0.144723	+	0.989472i	$0.453771\pi$
$480$	0	0
$481$	12.0484	0.549359
$482$	0	0
$483$	0.0566630	0.00257825
$484$	0	0
$485$	−1.34948	−0.0612769
$486$	0	0
$487$	−21.2210	−0.961617	−0.480809	−	0.876825i	$-0.659657\pi$
$487$	−21.2210	−0.961617	−0.480809	+	0.876825i	$0.659657\pi$
$488$	0	0
$489$	49.1328	2.22186
$490$	0	0
$491$	−37.0589	−1.67244	−0.836222	−	0.548390i	$-0.815241\pi$
$491$	−37.0589	−1.67244	−0.836222	+	0.548390i	$0.815241\pi$
$492$	0	0
$493$	−8.04125	−0.362160
$494$	0	0
$495$	−9.59649	−0.431330
$496$	0	0
$497$	4.25065	0.190668
$498$	0	0
$499$	−7.12480	−0.318950	−0.159475	−	0.987202i	$-0.550980\pi$
$499$	−7.12480	−0.318950	−0.159475	+	0.987202i	$0.550980\pi$
$500$	0	0
$501$	59.6777	2.66620
$502$	0	0
$503$	4.88102	0.217634	0.108817	−	0.994062i	$-0.465294\pi$
$503$	4.88102	0.217634	0.108817	+	0.994062i	$0.465294\pi$
$504$	0	0
$505$	−14.5891	−0.649206
$506$	0	0
$507$	−31.9998	−1.42116
$508$	0	0
$509$	−0.254610	−0.0112854	−0.00564270	−	0.999984i	$-0.501796\pi$
$509$	−0.254610	−0.0112854	−0.00564270	+	0.999984i	$0.501796\pi$
$510$	0	0
$511$	−2.78925	−0.123389
$512$	0	0
$513$	87.0658	3.84405
$514$	0	0
$515$	22.1292	0.975130
$516$	0	0
$517$	−4.99746	−0.219788
$518$	0	0
$519$	19.9813	0.877082
$520$	0	0
$521$	26.4908	1.16058	0.580291	−	0.814409i	$-0.302939\pi$
$521$	26.4908	1.16058	0.580291	+	0.814409i	$0.302939\pi$
$522$	0	0
$523$	−10.3065	−0.450673	−0.225336	−	0.974281i	$-0.572348\pi$
$523$	−10.3065	−0.450673	−0.225336	+	0.974281i	$0.572348\pi$
$524$	0	0
$525$	−0.321089	−0.0140135
$526$	0	0
$527$	10.5164	0.458102
$528$	0	0
$529$	−22.9991	−0.999961
$530$	0	0
$531$	84.1128	3.65019
$532$	0	0
$533$	15.7459	0.682029
$534$	0	0
$535$	2.58414	0.111722
$536$	0	0
$537$	22.1697	0.956694
$538$	0	0
$539$	−3.74550	−0.161330
$540$	0	0
$541$	−13.9736	−0.600772	−0.300386	−	0.953818i	$-0.597115\pi$
$541$	−13.9736	−0.600772	−0.300386	+	0.953818i	$0.597115\pi$
$542$	0	0
$543$	−13.7619	−0.590582
$544$	0	0
$545$	−30.2473	−1.29565
$546$	0	0
$547$	1.00000	0.0427569
$547$	1.00000	0.0427569
$548$	0	0
$549$	−56.2883	−2.40233
$550$	0	0
$551$	12.0724	0.514303
$552$	0	0
$553$	8.66773	0.368589
$554$	0	0
$555$	−50.2126	−2.13140
$556$	0	0
$557$	−7.56610	−0.320586	−0.160293	−	0.987069i	$-0.551244\pi$
$557$	−7.56610	−0.320586	−0.160293	+	0.987069i	$0.551244\pi$
$558$	0	0
$559$	9.96951	0.421665
$560$	0	0
$561$	−7.25259	−0.306204
$562$	0	0
$563$	−4.92242	−0.207455	−0.103728	−	0.994606i	$-0.533077\pi$
$563$	−4.92242	−0.207455	−0.103728	+	0.994606i	$0.533077\pi$
$564$	0	0
$565$	−6.67570	−0.280849
$566$	0	0
$567$	−14.5342	−0.610378
$568$	0	0
$569$	6.39281	0.268000	0.134000	−	0.990981i	$-0.457218\pi$
$569$	6.39281	0.268000	0.134000	+	0.990981i	$0.457218\pi$
$570$	0	0
$571$	17.6163	0.737220	0.368610	−	0.929584i	$-0.379834\pi$
$571$	17.6163	0.737220	0.368610	+	0.929584i	$0.379834\pi$
$572$	0	0
$573$	58.4656	2.44244
$574$	0	0
$575$	−0.00502548	−0.000209577 0
$576$	0	0
$577$	−19.7554	−0.822426	−0.411213	−	0.911539i	$-0.634895\pi$
$577$	−19.7554	−0.822426	−0.411213	+	0.911539i	$0.634895\pi$
$578$	0	0
$579$	10.8108	0.449282
$580$	0	0
$581$	−3.91252	−0.162319
$582$	0	0
$583$	6.66219	0.275920
$584$	0	0
$585$	−30.1420	−1.24622
$586$	0	0
$587$	23.1482	0.955428	0.477714	−	0.878515i	$-0.341465\pi$
$587$	23.1482	0.955428	0.477714	+	0.878515i	$0.341465\pi$
$588$	0	0
$589$	−15.7884	−0.650550
$590$	0	0
$591$	−51.8908	−2.13450
$592$	0	0
$593$	3.99710	0.164141	0.0820706	−	0.996627i	$-0.473847\pi$
$593$	3.99710	0.164141	0.0820706	+	0.996627i	$0.473847\pi$
$594$	0	0
$595$	−5.30903	−0.217649
$596$	0	0
$597$	47.8647	1.95897
$598$	0	0
$599$	29.3903	1.20086	0.600428	−	0.799679i	$-0.294997\pi$
$599$	29.3903	1.20086	0.600428	+	0.799679i	$0.294997\pi$
$600$	0	0
$601$	−34.0838	−1.39031	−0.695154	−	0.718861i	$-0.744664\pi$
$601$	−34.0838	−1.39031	−0.695154	+	0.718861i	$0.744664\pi$
$602$	0	0
$603$	−28.2815	−1.15171
$604$	0	0
$605$	24.2883	0.987459
$606$	0	0
$607$	16.0013	0.649473	0.324737	−	0.945804i	$-0.394724\pi$
$607$	16.0013	0.649473	0.324737	+	0.945804i	$0.394724\pi$
$608$	0	0
$609$	−3.84725	−0.155898
$610$	0	0
$611$	−15.6968	−0.635023
$612$	0	0
$613$	−48.6288	−1.96410	−0.982049	−	0.188623i	$-0.939597\pi$
$613$	−48.6288	−1.96410	−0.982049	+	0.188623i	$0.939597\pi$
$614$	0	0
$615$	−65.6222	−2.64614
$616$	0	0
$617$	−13.1578	−0.529712	−0.264856	−	0.964288i	$-0.585324\pi$
$617$	−13.1578	−0.529712	−0.264856	+	0.964288i	$0.585324\pi$
$618$	0	0
$619$	35.6049	1.43108	0.715540	−	0.698572i	$-0.246181\pi$
$619$	35.6049	1.43108	0.715540	+	0.698572i	$0.246181\pi$
$620$	0	0
$621$	−0.434265	−0.0174265
$622$	0	0
$623$	2.70443	0.108351
$624$	0	0
$625$	−25.8153	−1.03261
$626$	0	0
$627$	10.8884	0.434841
$628$	0	0
$629$	−27.1062	−1.08079
$630$	0	0
$631$	3.98773	0.158749	0.0793745	−	0.996845i	$-0.474708\pi$
$631$	3.98773	0.158749	0.0793745	+	0.996845i	$0.474708\pi$
$632$	0	0
$633$	30.2738	1.20328
$634$	0	0
$635$	−23.0860	−0.916141
$636$	0	0
$637$	−11.7644	−0.466122
$638$	0	0
$639$	−54.2941	−2.14784
$640$	0	0
$641$	−30.9516	−1.22252	−0.611258	−	0.791432i	$-0.709336\pi$
$641$	−30.9516	−1.22252	−0.611258	+	0.791432i	$0.709336\pi$
$642$	0	0
$643$	13.8181	0.544934	0.272467	−	0.962165i	$-0.412160\pi$
$643$	13.8181	0.544934	0.272467	+	0.962165i	$0.412160\pi$
$644$	0	0
$645$	−41.5487	−1.63598
$646$	0	0
$647$	−30.7237	−1.20788	−0.603938	−	0.797032i	$-0.706402\pi$
$647$	−30.7237	−1.20788	−0.603938	+	0.797032i	$0.706402\pi$
$648$	0	0
$649$	6.31157	0.247751
$650$	0	0
$651$	5.03146	0.197198
$652$	0	0
$653$	27.9296	1.09297	0.546484	−	0.837469i	$-0.315966\pi$
$653$	27.9296	1.09297	0.546484	+	0.837469i	$0.315966\pi$
$654$	0	0
$655$	−2.46525	−0.0963252
$656$	0	0
$657$	35.6274	1.38996
$658$	0	0
$659$	−5.40191	−0.210429	−0.105214	−	0.994450i	$-0.533553\pi$
$659$	−5.40191	−0.210429	−0.105214	+	0.994450i	$0.533553\pi$
$660$	0	0
$661$	−1.04994	−0.0408377	−0.0204189	−	0.999792i	$-0.506500\pi$
$661$	−1.04994	−0.0408377	−0.0204189	+	0.999792i	$0.506500\pi$
$662$	0	0
$663$	−22.7800	−0.884701
$664$	0	0
$665$	7.97052	0.309083
$666$	0	0
$667$	−0.0602146	−0.00233152
$668$	0	0
$669$	16.1266	0.623492
$670$	0	0
$671$	−4.22370	−0.163054
$672$	0	0
$673$	26.9770	1.03989	0.519943	−	0.854201i	$-0.325953\pi$
$673$	26.9770	1.03989	0.519943	+	0.854201i	$0.325953\pi$
$674$	0	0
$675$	2.46083	0.0947174
$676$	0	0
$677$	11.7287	0.450772	0.225386	−	0.974270i	$-0.427636\pi$
$677$	11.7287	0.450772	0.225386	+	0.974270i	$0.427636\pi$
$678$	0	0
$679$	−0.348538	−0.0133757
$680$	0	0
$681$	−42.2400	−1.61864
$682$	0	0
$683$	−19.2853	−0.737931	−0.368965	−	0.929443i	$-0.620288\pi$
$683$	−19.2853	−0.737931	−0.368965	+	0.929443i	$0.620288\pi$
$684$	0	0
$685$	−27.0367	−1.03302
$686$	0	0
$687$	−51.2911	−1.95688
$688$	0	0
$689$	20.9256	0.797201
$690$	0	0
$691$	4.50485	0.171373	0.0856863	−	0.996322i	$-0.472692\pi$
$691$	4.50485	0.171373	0.0856863	+	0.996322i	$0.472692\pi$
$692$	0	0
$693$	−2.47853	−0.0941517
$694$	0	0
$695$	−43.4759	−1.64913
$696$	0	0
$697$	−35.4247	−1.34181
$698$	0	0
$699$	38.2109	1.44527
$700$	0	0
$701$	23.3201	0.880787	0.440394	−	0.897805i	$-0.354839\pi$
$701$	23.3201	0.880787	0.440394	+	0.897805i	$0.354839\pi$
$702$	0	0
$703$	40.6948	1.53483
$704$	0	0
$705$	65.4175	2.46377
$706$	0	0
$707$	−3.76800	−0.141710
$708$	0	0
$709$	−15.8879	−0.596683	−0.298341	−	0.954459i	$-0.596433\pi$
$709$	−15.8879	−0.596683	−0.298341	+	0.954459i	$0.596433\pi$
$710$	0	0
$711$	−110.714	−4.15210
$712$	0	0
$713$	0.0787491	0.00294918
$714$	0	0
$715$	−2.26177	−0.0845853
$716$	0	0
$717$	9.55239	0.356741
$718$	0	0
$719$	8.94825	0.333714	0.166857	−	0.985981i	$-0.446638\pi$
$719$	8.94825	0.333714	0.166857	+	0.985981i	$0.446638\pi$
$720$	0	0
$721$	5.71542	0.212853
$722$	0	0
$723$	−37.8928	−1.40925
$724$	0	0
$725$	0.341215	0.0126724
$726$	0	0
$727$	19.1142	0.708905	0.354452	−	0.935074i	$-0.384667\pi$
$727$	19.1142	0.708905	0.354452	+	0.935074i	$0.384667\pi$
$728$	0	0
$729$	43.8883	1.62549
$730$	0	0
$731$	−22.4292	−0.829573
$732$	0	0
$733$	8.46101	0.312514	0.156257	−	0.987716i	$-0.450057\pi$
$733$	8.46101	0.312514	0.156257	+	0.987716i	$0.450057\pi$
$734$	0	0
$735$	49.0290	1.80846
$736$	0	0
$737$	−2.12216	−0.0781707
$738$	0	0
$739$	−0.611238	−0.0224848	−0.0112424	−	0.999937i	$-0.503579\pi$
$739$	−0.611238	−0.0224848	−0.0112424	+	0.999937i	$0.503579\pi$
$740$	0	0
$741$	34.1999	1.25636
$742$	0	0
$743$	−46.7180	−1.71392	−0.856958	−	0.515387i	$-0.827648\pi$
$743$	−46.7180	−1.71392	−0.856958	+	0.515387i	$0.827648\pi$
$744$	0	0
$745$	−40.8566	−1.49687
$746$	0	0
$747$	49.9751	1.82849
$748$	0	0
$749$	0.667418	0.0243869
$750$	0	0
$751$	−13.2159	−0.482257	−0.241128	−	0.970493i	$-0.577517\pi$
$751$	−13.2159	−0.482257	−0.241128	+	0.970493i	$0.577517\pi$
$752$	0	0
$753$	64.1979	2.33950
$754$	0	0
$755$	−4.90105	−0.178367
$756$	0	0
$757$	50.1147	1.82145	0.910725	−	0.413013i	$-0.135524\pi$
$757$	50.1147	1.82145	0.910725	+	0.413013i	$0.135524\pi$
$758$	0	0
$759$	−0.0543089	−0.00197129
$760$	0	0
$761$	−44.6658	−1.61913	−0.809567	−	0.587027i	$-0.800298\pi$
$761$	−44.6658	−1.61913	−0.809567	+	0.587027i	$0.800298\pi$
$762$	0	0
$763$	−7.81211	−0.282817
$764$	0	0
$765$	67.8129	2.45178
$766$	0	0
$767$	19.8243	0.715813
$768$	0	0
$769$	2.34044	0.0843983	0.0421992	−	0.999109i	$-0.486564\pi$
$769$	2.34044	0.0843983	0.0421992	+	0.999109i	$0.486564\pi$
$770$	0	0
$771$	55.1585	1.98648
$772$	0	0
$773$	−37.2217	−1.33877	−0.669386	−	0.742915i	$-0.733443\pi$
$773$	−37.2217	−1.33877	−0.669386	+	0.742915i	$0.733443\pi$
$774$	0	0
$775$	−0.446244	−0.0160295
$776$	0	0
$777$	−12.9687	−0.465248
$778$	0	0
$779$	53.1835	1.90550
$780$	0	0
$781$	−4.07406	−0.145781
$782$	0	0
$783$	29.4853	1.05372
$784$	0	0
$785$	9.39122	0.335187
$786$	0	0
$787$	−25.2383	−0.899647	−0.449823	−	0.893117i	$-0.648513\pi$
$787$	−25.2383	−0.899647	−0.449823	+	0.893117i	$0.648513\pi$
$788$	0	0
$789$	−12.0835	−0.430185
$790$	0	0
$791$	−1.72417	−0.0613043
$792$	0	0
$793$	−13.2664	−0.471104
$794$	0	0
$795$	−87.2090	−3.09298
$796$	0	0
$797$	−42.7479	−1.51421	−0.757104	−	0.653295i	$-0.773386\pi$
$797$	−42.7479	−1.51421	−0.757104	+	0.653295i	$0.773386\pi$
$798$	0	0
$799$	35.3142	1.24933
$800$	0	0
$801$	−34.5440	−1.22055
$802$	0	0
$803$	2.67337	0.0943412
$804$	0	0
$805$	−0.0397552	−0.00140119
$806$	0	0
$807$	28.4703	1.00220
$808$	0	0
$809$	−55.6137	−1.95527	−0.977637	−	0.210301i	$-0.932555\pi$
$809$	−55.6137	−1.95527	−0.977637	+	0.210301i	$0.932555\pi$
$810$	0	0
$811$	−46.9117	−1.64729	−0.823646	−	0.567104i	$-0.808064\pi$
$811$	−46.9117	−1.64729	−0.823646	+	0.567104i	$0.808064\pi$
$812$	0	0
$813$	−21.6041	−0.757690
$814$	0	0
$815$	−34.4719	−1.20750
$816$	0	0
$817$	33.6732	1.17808
$818$	0	0
$819$	−7.78494	−0.272028
$820$	0	0
$821$	−25.3283	−0.883965	−0.441982	−	0.897024i	$-0.645725\pi$
$821$	−25.3283	−0.883965	−0.441982	+	0.897024i	$0.645725\pi$
$822$	0	0
$823$	46.7507	1.62963	0.814813	−	0.579724i	$-0.196840\pi$
$823$	46.7507	1.62963	0.814813	+	0.579724i	$0.196840\pi$
$824$	0	0
$825$	0.307750	0.0107145
$826$	0	0
$827$	−5.41638	−0.188346	−0.0941729	−	0.995556i	$-0.530021\pi$
$827$	−5.41638	−0.188346	−0.0941729	+	0.995556i	$0.530021\pi$
$828$	0	0
$829$	49.3103	1.71262	0.856309	−	0.516463i	$-0.172752\pi$
$829$	49.3103	1.71262	0.856309	+	0.516463i	$0.172752\pi$
$830$	0	0
$831$	−104.535	−3.62629
$832$	0	0
$833$	26.4673	0.917036
$834$	0	0
$835$	−41.8703	−1.44898
$836$	0	0
$837$	−38.5611	−1.33287
$838$	0	0
$839$	−45.9938	−1.58788	−0.793941	−	0.607995i	$-0.791974\pi$
$839$	−45.9938	−1.58788	−0.793941	+	0.607995i	$0.791974\pi$
$840$	0	0
$841$	−24.9116	−0.859021
$842$	0	0
$843$	38.5027	1.32610
$844$	0	0
$845$	22.4513	0.772347
$846$	0	0
$847$	6.27306	0.215545
$848$	0	0
$849$	−73.9268	−2.53716
$850$	0	0
$851$	−0.202977	−0.00695795
$852$	0	0
$853$	−21.5129	−0.736589	−0.368295	−	0.929709i	$-0.620058\pi$
$853$	−21.5129	−0.736589	−0.368295	+	0.929709i	$0.620058\pi$
$854$	0	0
$855$	−101.808	−3.48177
$856$	0	0
$857$	−54.1519	−1.84979	−0.924897	−	0.380217i	$-0.875849\pi$
$857$	−54.1519	−1.84979	−0.924897	+	0.380217i	$0.875849\pi$
$858$	0	0
$859$	−16.5943	−0.566190	−0.283095	−	0.959092i	$-0.591361\pi$
$859$	−16.5943	−0.566190	−0.283095	+	0.959092i	$0.591361\pi$
$860$	0	0
$861$	−16.9486	−0.577606
$862$	0	0
$863$	−34.3208	−1.16830	−0.584148	−	0.811648i	$-0.698571\pi$
$863$	−34.3208	−1.16830	−0.584148	+	0.811648i	$0.698571\pi$
$864$	0	0
$865$	−14.0190	−0.476661
$866$	0	0
$867$	−3.83695	−0.130310
$868$	0	0
$869$	−8.30763	−0.281817
$870$	0	0
$871$	−6.66558	−0.225855
$872$	0	0
$873$	4.45191	0.150674
$874$	0	0
$875$	−6.44952	−0.218034
$876$	0	0
$877$	3.56028	0.120222	0.0601110	−	0.998192i	$-0.480855\pi$
$877$	3.56028	0.120222	0.0601110	+	0.998192i	$0.480855\pi$
$878$	0	0
$879$	−3.88917	−0.131179
$880$	0	0
$881$	25.4637	0.857894	0.428947	−	0.903330i	$-0.358885\pi$
$881$	25.4637	0.857894	0.428947	+	0.903330i	$0.358885\pi$
$882$	0	0
$883$	43.7365	1.47185	0.735925	−	0.677063i	$-0.236748\pi$
$883$	43.7365	1.47185	0.735925	+	0.677063i	$0.236748\pi$
$884$	0	0
$885$	−82.6193	−2.77722
$886$	0	0
$887$	4.63265	0.155549	0.0777745	−	0.996971i	$-0.475219\pi$
$887$	4.63265	0.155549	0.0777745	+	0.996971i	$0.475219\pi$
$888$	0	0
$889$	−5.96255	−0.199977
$890$	0	0
$891$	13.9304	0.466685
$892$	0	0
$893$	−53.0177	−1.77417
$894$	0	0
$895$	−15.5544	−0.519928
$896$	0	0
$897$	−0.170581	−0.00569555
$898$	0	0
$899$	−5.34683	−0.178327
$900$	0	0
$901$	−47.0779	−1.56839
$902$	0	0
$903$	−10.7310	−0.357105
$904$	0	0
$905$	9.65548	0.320959
$906$	0	0
$907$	35.0844	1.16496	0.582478	−	0.812846i	$-0.302083\pi$
$907$	35.0844	1.16496	0.582478	+	0.812846i	$0.302083\pi$
$908$	0	0
$909$	48.1291	1.59634
$910$	0	0
$911$	12.3081	0.407785	0.203893	−	0.978993i	$-0.434641\pi$
$911$	12.3081	0.407785	0.203893	+	0.978993i	$0.434641\pi$
$912$	0	0
$913$	3.74998	0.124106
$914$	0	0
$915$	55.2888	1.82779
$916$	0	0
$917$	−0.636712	−0.0210261
$918$	0	0
$919$	14.0514	0.463515	0.231757	−	0.972774i	$-0.425552\pi$
$919$	14.0514	0.463515	0.231757	+	0.972774i	$0.425552\pi$
$920$	0	0
$921$	−41.4665	−1.36637
$922$	0	0
$923$	−12.7964	−0.421199
$924$	0	0
$925$	1.15020	0.0378183
$926$	0	0
$927$	−73.0038	−2.39776
$928$	0	0
$929$	10.6938	0.350853	0.175427	−	0.984493i	$-0.443870\pi$
$929$	10.6938	0.350853	0.175427	+	0.984493i	$0.443870\pi$
$930$	0	0
$931$	−39.7356	−1.30228
$932$	0	0
$933$	−56.3146	−1.84366
$934$	0	0
$935$	5.08847	0.166411
$936$	0	0
$937$	−1.62113	−0.0529601	−0.0264800	−	0.999649i	$-0.508430\pi$
$937$	−1.62113	−0.0529601	−0.0264800	+	0.999649i	$0.508430\pi$
$938$	0	0
$939$	−19.1815	−0.625966
$940$	0	0
$941$	21.9839	0.716654	0.358327	−	0.933596i	$-0.383347\pi$
$941$	21.9839	0.716654	0.358327	+	0.933596i	$0.383347\pi$
$942$	0	0
$943$	−0.265268	−0.00863831
$944$	0	0
$945$	19.4669	0.633260
$946$	0	0
$947$	−11.7706	−0.382492	−0.191246	−	0.981542i	$-0.561253\pi$
$947$	−11.7706	−0.382492	−0.191246	+	0.981542i	$0.561253\pi$
$948$	0	0
$949$	8.39691	0.272575
$950$	0	0
$951$	−54.3239	−1.76157
$952$	0	0
$953$	10.6036	0.343485	0.171743	−	0.985142i	$-0.445060\pi$
$953$	10.6036	0.343485	0.171743	+	0.985142i	$0.445060\pi$
$954$	0	0
$955$	−41.0199	−1.32737
$956$	0	0
$957$	3.68742	0.119197
$958$	0	0
$959$	−6.98291	−0.225490
$960$	0	0
$961$	−24.0074	−0.774431
$962$	0	0
$963$	−8.52501	−0.274715
$964$	0	0
$965$	−7.58495	−0.244168
$966$	0	0
$967$	7.34001	0.236039	0.118019	−	0.993011i	$-0.462346\pi$
$967$	7.34001	0.236039	0.118019	+	0.993011i	$0.462346\pi$
$968$	0	0
$969$	−76.9421	−2.47173
$970$	0	0
$971$	46.6664	1.49760	0.748799	−	0.662797i	$-0.230631\pi$
$971$	46.6664	1.49760	0.748799	+	0.662797i	$0.230631\pi$
$972$	0	0
$973$	−11.2287	−0.359977
$974$	0	0
$975$	0.966625	0.0309568
$976$	0	0
$977$	21.3732	0.683789	0.341895	−	0.939738i	$-0.388931\pi$
$977$	21.3732	0.683789	0.341895	+	0.939738i	$0.388931\pi$
$978$	0	0
$979$	−2.59208	−0.0828432
$980$	0	0
$981$	99.7850	3.18589
$982$	0	0
$983$	−14.4017	−0.459344	−0.229672	−	0.973268i	$-0.573765\pi$
$983$	−14.4017	−0.459344	−0.229672	+	0.973268i	$0.573765\pi$
$984$	0	0
$985$	36.4070	1.16002
$986$	0	0
$987$	16.8957	0.537796
$988$	0	0
$989$	−0.167954	−0.00534064
$990$	0	0
$991$	6.12762	0.194650	0.0973251	−	0.995253i	$-0.468971\pi$
$991$	6.12762	0.194650	0.0973251	+	0.995253i	$0.468971\pi$
$992$	0	0
$993$	−93.4478	−2.96548
$994$	0	0
$995$	−33.5822	−1.06463
$996$	0	0
$997$	16.8290	0.532980	0.266490	−	0.963838i	$-0.414136\pi$
$997$	16.8290	0.532980	0.266490	+	0.963838i	$0.414136\pi$
$998$	0	0
$999$	99.3918	3.14462

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8752.2.a.x.1.29		31
4.3	odd	2		4376.2.a.i.1.3	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4376.2.a.i.1.3	✓	31	4.3	odd	2
8752.2.a.x.1.29		31	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8752 = 2^{4} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8752.a (trivial)

\(f(q)\)	\(=\)	\(q+3.24040 q^{3} -2.27349 q^{5} -0.587186 q^{7} +7.50019 q^{9} +O(q^{10})\)	\(q+3.24040 q^{3} -2.27349 q^{5} -0.587186 q^{7} +7.50019 q^{9} +0.562791 q^{11} +1.76770 q^{13} -7.36701 q^{15} -3.97692 q^{17} +5.97060 q^{19} -1.90272 q^{21} -0.0297800 q^{23} +0.168753 q^{25} +14.5824 q^{27} +2.02198 q^{29} -2.64436 q^{31} +1.82367 q^{33} +1.33496 q^{35} +6.81586 q^{37} +5.72804 q^{39} +8.90757 q^{41} +5.63983 q^{43} -17.0516 q^{45} -8.87979 q^{47} -6.65521 q^{49} -12.8868 q^{51} +11.8378 q^{53} -1.27950 q^{55} +19.3471 q^{57} +11.2148 q^{59} -7.50492 q^{61} -4.40400 q^{63} -4.01884 q^{65} -3.77078 q^{67} -0.0964993 q^{69} -7.23903 q^{71} +4.75020 q^{73} +0.546828 q^{75} -0.330463 q^{77} -14.7615 q^{79} +24.7523 q^{81} +6.66318 q^{83} +9.04149 q^{85} +6.55202 q^{87} -4.60576 q^{89} -1.03797 q^{91} -8.56877 q^{93} -13.5741 q^{95} +0.593574 q^{97} +4.22104 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 8 q^{3} - 11 q^{5} + 10 q^{7} + 39 q^{9}+O(q^{10}) \) 31 * q + 8 * q^3 - 11 * q^5 + 10 * q^7 + 39 * q^9	\( 31 q + 8 q^{3} - 11 q^{5} + 10 q^{7} + 39 q^{9} + 11 q^{11} - 20 q^{13} + 11 q^{15} + 26 q^{17} + 32 q^{19} - 23 q^{21} + 29 q^{23} + 26 q^{25} + 53 q^{27} - 6 q^{29} + 53 q^{31} + 8 q^{33} + 13 q^{35} - 26 q^{37} + 24 q^{39} - 6 q^{41} + 11 q^{43} - 12 q^{45} + 41 q^{47} + 29 q^{49} + 35 q^{51} - 17 q^{53} + 10 q^{55} + 9 q^{57} + 27 q^{59} - 62 q^{61} + 34 q^{63} + 4 q^{65} + 28 q^{67} - 20 q^{69} + 39 q^{71} + 16 q^{73} + 36 q^{75} - 19 q^{77} + 39 q^{79} + 35 q^{81} + 32 q^{83} - 58 q^{85} + 38 q^{87} + 61 q^{89} + 25 q^{91} - q^{93} + 12 q^{95} + 85 q^{99}+O(q^{100}) \) 31 * q + 8 * q^3 - 11 * q^5 + 10 * q^7 + 39 * q^9 + 11 * q^11 - 20 * q^13 + 11 * q^15 + 26 * q^17 + 32 * q^19 - 23 * q^21 + 29 * q^23 + 26 * q^25 + 53 * q^27 - 6 * q^29 + 53 * q^31 + 8 * q^33 + 13 * q^35 - 26 * q^37 + 24 * q^39 - 6 * q^41 + 11 * q^43 - 12 * q^45 + 41 * q^47 + 29 * q^49 + 35 * q^51 - 17 * q^53 + 10 * q^55 + 9 * q^57 + 27 * q^59 - 62 * q^61 + 34 * q^63 + 4 * q^65 + 28 * q^67 - 20 * q^69 + 39 * q^71 + 16 * q^73 + 36 * q^75 - 19 * q^77 + 39 * q^79 + 35 * q^81 + 32 * q^83 - 58 * q^85 + 38 * q^87 + 61 * q^89 + 25 * q^91 - q^93 + 12 * q^95 + 85 * q^99

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