LMFDB - Embedded newform 3249.2.a.bc.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3249,2,Mod(1,3249)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3249.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3249, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3249 = 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3249.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,2,4,0,-8,0,0,0,10,0,0,0,0,-14,8,0,0,12,0,0,6,0,4,-10,0, 6,0,0,0,0,0,0,12,0,0,0,0,0,0,0,6,0,0,0,28,0,8,0,0,0,0,0,0,0,0,10,0,0,-20, 10,0,8,0,0,0,24,0,0,0,0,36,-10,0,0,-30,0,0,-24,0,30,4,0,48,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$25.9433956167$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 361)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.17557$ of defining polynomial
Character	$\chi$	$=$	3249.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17557 q^{2} -0.618034 q^{4} -1.23607 q^{5} -4.23607 q^{7} -3.07768 q^{8} -1.45309 q^{10} +3.61803 q^{11} -3.07768 q^{13} -4.97980 q^{14} -2.38197 q^{16} -2.47214 q^{17} +0.763932 q^{20} +4.25325 q^{22} +2.61803 q^{23} -3.47214 q^{25} -3.61803 q^{26} +2.61803 q^{28} -2.62866 q^{29} +1.17557 q^{31} +3.35520 q^{32} -2.90617 q^{34} +5.23607 q^{35} +6.43288 q^{37} +3.80423 q^{40} +9.23305 q^{41} -6.32624 q^{43} -2.23607 q^{44} +3.07768 q^{46} +2.52786 q^{47} +10.9443 q^{49} -4.08174 q^{50} +1.90211 q^{52} +0.449028 q^{53} -4.47214 q^{55} +13.0373 q^{56} -3.09017 q^{58} -2.80017 q^{59} -0.527864 q^{61} +1.38197 q^{62} +8.70820 q^{64} +3.80423 q^{65} +0.171513 q^{67} +1.52786 q^{68} +6.15537 q^{70} +12.1392 q^{71} +9.00000 q^{73} +7.56231 q^{74} -15.3262 q^{77} +7.60845 q^{79} +2.94427 q^{80} +10.8541 q^{82} -1.23607 q^{83} +3.05573 q^{85} -7.43694 q^{86} -11.1352 q^{88} -17.7396 q^{89} +13.0373 q^{91} -1.61803 q^{92} +2.97168 q^{94} +15.6659 q^{97} +12.8658 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 4 q^{5} - 8 q^{7} + 10 q^{11} - 14 q^{16} + 8 q^{17} + 12 q^{20} + 6 q^{23} + 4 q^{25} - 10 q^{26} + 6 q^{28} + 12 q^{35} + 6 q^{43} + 28 q^{47} + 8 q^{49} + 10 q^{58} - 20 q^{61} + 10 q^{62}+ \cdots - 2 q^{92}+O(q^{100}) $ 4 * q + 2 * q^4 + 4 * q^5 - 8 * q^7 + 10 * q^11 - 14 * q^16 + 8 * q^17 + 12 * q^20 + 6 * q^23 + 4 * q^25 - 10 * q^26 + 6 * q^28 + 12 * q^35 + 6 * q^43 + 28 * q^47 + 8 * q^49 + 10 * q^58 - 20 * q^61 + 10 * q^62 + 8 * q^64 + 24 * q^68 + 36 * q^73 - 10 * q^74 - 30 * q^77 - 24 * q^80 + 30 * q^82 + 4 * q^83 + 48 * q^85 - 2 * q^92

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 $

$2$	1.17557	0.831254	0.415627	−	0.909535i	$-0.363562\pi$
$2$	1.17557	0.831254	0.415627	+	0.909535i	$0.363562\pi$
$3$	0	0
$3$	0	0
$4$	−0.618034	−0.309017
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	−4.23607	−1.60108	−0.800542	−	0.599277i	$-0.795455\pi$
$7$	−4.23607	−1.60108	−0.800542	+	0.599277i	$0.795455\pi$
$8$	−3.07768	−1.08813
$9$	0	0
$10$	−1.45309	−0.459506
$11$	3.61803	1.09088	0.545439	−	0.838150i	$-0.316363\pi$
$11$	3.61803	1.09088	0.545439	+	0.838150i	$0.316363\pi$
$12$	0	0
$13$	−3.07768	−0.853596	−0.426798	−	0.904347i	$-0.640358\pi$
$13$	−3.07768	−0.853596	−0.426798	+	0.904347i	$0.640358\pi$
$14$	−4.97980	−1.33091
$15$	0	0
$16$	−2.38197	−0.595492
$17$	−2.47214	−0.599581	−0.299791	−	0.954005i	$-0.596917\pi$
$17$	−2.47214	−0.599581	−0.299791	+	0.954005i	$0.596917\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0.763932	0.170820
$21$	0	0
$22$	4.25325	0.906797
$23$	2.61803	0.545898	0.272949	−	0.962029i	$-0.412001\pi$
$23$	2.61803	0.545898	0.272949	+	0.962029i	$0.412001\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	−3.61803	−0.709555
$27$	0	0
$28$	2.61803	0.494762
$29$	−2.62866	−0.488129	−0.244065	−	0.969759i	$-0.578481\pi$
$29$	−2.62866	−0.488129	−0.244065	+	0.969759i	$0.578481\pi$
$30$	0	0
$31$	1.17557	0.211139	0.105569	−	0.994412i	$-0.466333\pi$
$31$	1.17557	0.211139	0.105569	+	0.994412i	$0.466333\pi$
$32$	3.35520	0.593121
$33$	0	0
$34$	−2.90617	−0.498404
$35$	5.23607	0.885057
$36$	0	0
$37$	6.43288	1.05756	0.528780	−	0.848759i	$-0.322650\pi$
$37$	6.43288	1.05756	0.528780	+	0.848759i	$0.322650\pi$
$38$	0	0
$39$	0	0
$40$	3.80423	0.601501
$41$	9.23305	1.44196	0.720980	−	0.692956i	$-0.243692\pi$
$41$	9.23305	1.44196	0.720980	+	0.692956i	$0.243692\pi$
$42$	0	0
$43$	−6.32624	−0.964742	−0.482371	−	0.875967i	$-0.660224\pi$
$43$	−6.32624	−0.964742	−0.482371	+	0.875967i	$0.660224\pi$
$44$	−2.23607	−0.337100
$45$	0	0
$46$	3.07768	0.453780
$47$	2.52786	0.368727	0.184363	−	0.982858i	$-0.440978\pi$
$47$	2.52786	0.368727	0.184363	+	0.982858i	$0.440978\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	−4.08174	−0.577245
$51$	0	0
$52$	1.90211	0.263776
$53$	0.449028	0.0616787	0.0308394	−	0.999524i	$-0.490182\pi$
$53$	0.449028	0.0616787	0.0308394	+	0.999524i	$0.490182\pi$
$54$	0	0
$55$	−4.47214	−0.603023
$56$	13.0373	1.74218
$57$	0	0
$58$	−3.09017	−0.405759
$59$	−2.80017	−0.364551	−0.182275	−	0.983248i	$-0.558346\pi$
$59$	−2.80017	−0.364551	−0.182275	+	0.983248i	$0.558346\pi$
$60$	0	0
$61$	−0.527864	−0.0675861	−0.0337930	−	0.999429i	$-0.510759\pi$
$61$	−0.527864	−0.0675861	−0.0337930	+	0.999429i	$0.510759\pi$
$62$	1.38197	0.175510
$63$	0	0
$64$	8.70820	1.08853
$65$	3.80423	0.471856
$66$	0	0
$67$	0.171513	0.0209537	0.0104768	−	0.999945i	$-0.496665\pi$
$67$	0.171513	0.0209537	0.0104768	+	0.999945i	$0.496665\pi$
$68$	1.52786	0.185281
$69$	0	0
$70$	6.15537	0.735707
$71$	12.1392	1.44066	0.720330	−	0.693631i	$-0.243990\pi$
$71$	12.1392	1.44066	0.720330	+	0.693631i	$0.243990\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	7.56231	0.879100
$75$	0	0
$76$	0	0
$77$	−15.3262	−1.74659
$78$	0	0
$79$	7.60845	0.856018	0.428009	−	0.903775i	$-0.359215\pi$
$79$	7.60845	0.856018	0.428009	+	0.903775i	$0.359215\pi$
$80$	2.94427	0.329180
$81$	0	0
$82$	10.8541	1.19864
$83$	−1.23607	−0.135676	−0.0678380	−	0.997696i	$-0.521610\pi$
$83$	−1.23607	−0.135676	−0.0678380	+	0.997696i	$0.521610\pi$
$84$	0	0
$85$	3.05573	0.331440
$86$	−7.43694	−0.801946
$87$	0	0
$88$	−11.1352	−1.18701
$89$	−17.7396	−1.88039	−0.940195	−	0.340637i	$-0.889357\pi$
$89$	−17.7396	−1.88039	−0.940195	+	0.340637i	$0.889357\pi$
$90$	0	0
$91$	13.0373	1.36668
$92$	−1.61803	−0.168692
$93$	0	0
$94$	2.97168	0.306506
$95$	0	0
$96$	0	0
$97$	15.6659	1.59063	0.795317	−	0.606193i	$-0.207304\pi$
$97$	15.6659	1.59063	0.795317	+	0.606193i	$0.207304\pi$
$98$	12.8658	1.29964
$99$	0	0
$100$	2.14590	0.214590
$101$	13.9443	1.38751	0.693753	−	0.720213i	$-0.255956\pi$
$101$	13.9443	1.38751	0.693753	+	0.720213i	$0.255956\pi$
$102$	0	0
$103$	−1.00406	−0.0989327	−0.0494663	−	0.998776i	$-0.515752\pi$
$103$	−1.00406	−0.0989327	−0.0494663	+	0.998776i	$0.515752\pi$
$104$	9.47214	0.928819
$105$	0	0
$106$	0.527864	0.0512707
$107$	−11.5842	−1.11989	−0.559943	−	0.828531i	$-0.689177\pi$
$107$	−11.5842	−1.11989	−0.559943	+	0.828531i	$0.689177\pi$
$108$	0	0
$109$	0.726543	0.0695902	0.0347951	−	0.999394i	$-0.488922\pi$
$109$	0.726543	0.0695902	0.0347951	+	0.999394i	$0.488922\pi$
$110$	−5.25731	−0.501265
$111$	0	0
$112$	10.0902	0.953431
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	−3.23607	−0.301765
$116$	1.62460	0.150840
$117$	0	0
$118$	−3.29180	−0.303034
$119$	10.4721	0.959979
$120$	0	0
$121$	2.09017	0.190015
$122$	−0.620541	−0.0561812
$123$	0	0
$124$	−0.726543	−0.0652454
$125$	10.4721	0.936656
$126$	0	0
$127$	−15.9434	−1.41475	−0.707376	−	0.706837i	$-0.750121\pi$
$127$	−15.9434	−1.41475	−0.707376	+	0.706837i	$0.750121\pi$
$128$	3.52671	0.311720
$129$	0	0
$130$	4.47214	0.392232
$131$	5.09017	0.444730	0.222365	−	0.974963i	$-0.428622\pi$
$131$	5.09017	0.444730	0.222365	+	0.974963i	$0.428622\pi$
$132$	0	0
$133$	0	0
$134$	0.201626	0.0174178
$135$	0	0
$136$	7.60845	0.652419
$137$	4.70820	0.402249	0.201125	−	0.979566i	$-0.435540\pi$
$137$	4.70820	0.402249	0.201125	+	0.979566i	$0.435540\pi$
$138$	0	0
$139$	−0.909830	−0.0771708	−0.0385854	−	0.999255i	$-0.512285\pi$
$139$	−0.909830	−0.0771708	−0.0385854	+	0.999255i	$0.512285\pi$
$140$	−3.23607	−0.273498
$141$	0	0
$142$	14.2705	1.19755
$143$	−11.1352	−0.931169
$144$	0	0
$145$	3.24920	0.269831
$146$	10.5801	0.875618
$147$	0	0
$148$	−3.97574	−0.326804
$149$	23.0344	1.88706	0.943528	−	0.331294i	$-0.107485\pi$
$149$	23.0344	1.88706	0.943528	+	0.331294i	$0.107485\pi$
$150$	0	0
$151$	2.45714	0.199959	0.0999797	−	0.994989i	$-0.468122\pi$
$151$	2.45714	0.199959	0.0999797	+	0.994989i	$0.468122\pi$
$152$	0	0
$153$	0	0
$154$	−18.0171	−1.45186
$155$	−1.45309	−0.116715
$156$	0	0
$157$	15.5623	1.24201	0.621004	−	0.783808i	$-0.286725\pi$
$157$	15.5623	1.24201	0.621004	+	0.783808i	$0.286725\pi$
$158$	8.94427	0.711568
$159$	0	0
$160$	−4.14725	−0.327869
$161$	−11.0902	−0.874028
$162$	0	0
$163$	6.52786	0.511302	0.255651	−	0.966769i	$-0.417710\pi$
$163$	6.52786	0.511302	0.255651	+	0.966769i	$0.417710\pi$
$164$	−5.70634	−0.445590
$165$	0	0
$166$	−1.45309	−0.112781
$167$	6.88191	0.532538	0.266269	−	0.963899i	$-0.414209\pi$
$167$	6.88191	0.532538	0.266269	+	0.963899i	$0.414209\pi$
$168$	0	0
$169$	−3.52786	−0.271374
$170$	3.59222	0.275511
$171$	0	0
$172$	3.90983	0.298122
$173$	15.5599	1.18300	0.591500	−	0.806305i	$-0.298536\pi$
$173$	15.5599	1.18300	0.591500	+	0.806305i	$0.298536\pi$
$174$	0	0
$175$	14.7082	1.11184
$176$	−8.61803	−0.649609
$177$	0	0
$178$	−20.8541	−1.56308
$179$	−18.2946	−1.36740	−0.683701	−	0.729762i	$-0.739631\pi$
$179$	−18.2946	−1.36740	−0.683701	+	0.729762i	$0.739631\pi$
$180$	0	0
$181$	−24.6215	−1.83010	−0.915050	−	0.403341i	$-0.867849\pi$
$181$	−24.6215	−1.83010	−0.915050	+	0.403341i	$0.867849\pi$
$182$	15.3262	1.13606
$183$	0	0
$184$	−8.05748	−0.594005
$185$	−7.95148	−0.584604
$186$	0	0
$187$	−8.94427	−0.654070
$188$	−1.56231	−0.113943
$189$	0	0
$190$	0	0
$191$	17.0000	1.23008	0.615038	−	0.788497i	$-0.289140\pi$
$191$	17.0000	1.23008	0.615038	+	0.788497i	$0.289140\pi$
$192$	0	0
$193$	−5.25731	−0.378430	−0.189215	−	0.981936i	$-0.560594\pi$
$193$	−5.25731	−0.378430	−0.189215	+	0.981936i	$0.560594\pi$
$194$	18.4164	1.32222
$195$	0	0
$196$	−6.76393	−0.483138
$197$	−2.52786	−0.180103	−0.0900514	−	0.995937i	$-0.528703\pi$
$197$	−2.52786	−0.180103	−0.0900514	+	0.995937i	$0.528703\pi$
$198$	0	0
$199$	−18.9443	−1.34292	−0.671462	−	0.741039i	$-0.734333\pi$
$199$	−18.9443	−1.34292	−0.671462	+	0.741039i	$0.734333\pi$
$200$	10.6861	0.755624
$201$	0	0
$202$	16.3925	1.15337
$203$	11.1352	0.781535
$204$	0	0
$205$	−11.4127	−0.797096
$206$	−1.18034	−0.0822382
$207$	0	0
$208$	7.33094	0.508309
$209$	0	0
$210$	0	0
$211$	−18.0171	−1.24035	−0.620173	−	0.784465i	$-0.712938\pi$
$211$	−18.0171	−1.24035	−0.620173	+	0.784465i	$0.712938\pi$
$212$	−0.277515	−0.0190598
$213$	0	0
$214$	−13.6180	−0.930910
$215$	7.81966	0.533296
$216$	0	0
$217$	−4.97980	−0.338051
$218$	0.854102	0.0578471
$219$	0	0
$220$	2.76393	0.186344
$221$	7.60845	0.511800
$222$	0	0
$223$	13.5923	0.910208	0.455104	−	0.890438i	$-0.349602\pi$
$223$	13.5923	0.910208	0.455104	+	0.890438i	$0.349602\pi$
$224$	−14.2128	−0.949636
$225$	0	0
$226$	0	0
$227$	29.7073	1.97174	0.985870	−	0.167511i	$-0.0535729\pi$
$227$	29.7073	1.97174	0.985870	+	0.167511i	$0.0535729\pi$
$228$	0	0
$229$	4.09017	0.270286	0.135143	−	0.990826i	$-0.456851\pi$
$229$	4.09017	0.270286	0.135143	+	0.990826i	$0.456851\pi$
$230$	−3.80423	−0.250843
$231$	0	0
$232$	8.09017	0.531146
$233$	−13.4721	−0.882589	−0.441294	−	0.897362i	$-0.645481\pi$
$233$	−13.4721	−0.882589	−0.441294	+	0.897362i	$0.645481\pi$
$234$	0	0
$235$	−3.12461	−0.203827
$236$	1.73060	0.112652
$237$	0	0
$238$	12.3107	0.797986
$239$	15.8541	1.02552	0.512758	−	0.858533i	$-0.328624\pi$
$239$	15.8541	1.02552	0.512758	+	0.858533i	$0.328624\pi$
$240$	0	0
$241$	22.6538	1.45926	0.729631	−	0.683841i	$-0.239692\pi$
$241$	22.6538	1.45926	0.729631	+	0.683841i	$0.239692\pi$
$242$	2.45714	0.157951
$243$	0	0
$244$	0.326238	0.0208852
$245$	−13.5279	−0.864264
$246$	0	0
$247$	0	0
$248$	−3.61803	−0.229745
$249$	0	0
$250$	12.3107	0.778599
$251$	−10.4164	−0.657478	−0.328739	−	0.944421i	$-0.606624\pi$
$251$	−10.4164	−0.657478	−0.328739	+	0.944421i	$0.606624\pi$
$252$	0	0
$253$	9.47214	0.595508
$254$	−18.7426	−1.17602
$255$	0	0
$256$	−13.2705	−0.829407
$257$	−9.06154	−0.565243	−0.282622	−	0.959231i	$-0.591204\pi$
$257$	−9.06154	−0.565243	−0.282622	+	0.959231i	$0.591204\pi$
$258$	0	0
$259$	−27.2501	−1.69324
$260$	−2.35114	−0.145812
$261$	0	0
$262$	5.98385	0.369684
$263$	−11.8885	−0.733079	−0.366540	−	0.930402i	$-0.619458\pi$
$263$	−11.8885	−0.733079	−0.366540	+	0.930402i	$0.619458\pi$
$264$	0	0
$265$	−0.555029	−0.0340952
$266$	0	0
$267$	0	0
$268$	−0.106001	−0.00647505
$269$	7.71445	0.470359	0.235179	−	0.971952i	$-0.424432\pi$
$269$	7.71445	0.470359	0.235179	+	0.971952i	$0.424432\pi$
$270$	0	0
$271$	14.2705	0.866872	0.433436	−	0.901184i	$-0.357301\pi$
$271$	14.2705	0.866872	0.433436	+	0.901184i	$0.357301\pi$
$272$	5.88854	0.357045
$273$	0	0
$274$	5.53483	0.334371
$275$	−12.5623	−0.757536
$276$	0	0
$277$	−19.2361	−1.15578	−0.577892	−	0.816113i	$-0.696124\pi$
$277$	−19.2361	−1.15578	−0.577892	+	0.816113i	$0.696124\pi$
$278$	−1.06957	−0.0641485
$279$	0	0
$280$	−16.1150	−0.963053
$281$	4.42477	0.263959	0.131980	−	0.991252i	$-0.457867\pi$
$281$	4.42477	0.263959	0.131980	+	0.991252i	$0.457867\pi$
$282$	0	0
$283$	−0.909830	−0.0540838	−0.0270419	−	0.999634i	$-0.508609\pi$
$283$	−0.909830	−0.0540838	−0.0270419	+	0.999634i	$0.508609\pi$
$284$	−7.50245	−0.445189
$285$	0	0
$286$	−13.0902	−0.774038
$287$	−39.1118	−2.30870
$288$	0	0
$289$	−10.8885	−0.640503
$290$	3.81966	0.224298
$291$	0	0
$292$	−5.56231	−0.325509
$293$	−24.7275	−1.44459	−0.722297	−	0.691583i	$-0.756914\pi$
$293$	−24.7275	−1.44459	−0.722297	+	0.691583i	$0.756914\pi$
$294$	0	0
$295$	3.46120	0.201519
$296$	−19.7984	−1.15076
$297$	0	0
$298$	27.0786	1.56862
$299$	−8.05748	−0.465976
$300$	0	0
$301$	26.7984	1.54463
$302$	2.88854	0.166217
$303$	0	0
$304$	0	0
$305$	0.652476	0.0373607
$306$	0	0
$307$	−27.5932	−1.57482	−0.787412	−	0.616427i	$-0.788580\pi$
$307$	−27.5932	−1.57482	−0.787412	+	0.616427i	$0.788580\pi$
$308$	9.47214	0.539725
$309$	0	0
$310$	−1.70820	−0.0970195
$311$	5.52786	0.313456	0.156728	−	0.987642i	$-0.449905\pi$
$311$	5.52786	0.313456	0.156728	+	0.987642i	$0.449905\pi$
$312$	0	0
$313$	7.61803	0.430597	0.215298	−	0.976548i	$-0.430928\pi$
$313$	7.61803	0.430597	0.215298	+	0.976548i	$0.430928\pi$
$314$	18.2946	1.03242
$315$	0	0
$316$	−4.70228	−0.264524
$317$	−8.84953	−0.497039	−0.248520	−	0.968627i	$-0.579944\pi$
$317$	−8.84953	−0.497039	−0.248520	+	0.968627i	$0.579944\pi$
$318$	0	0
$319$	−9.51057	−0.532489
$320$	−10.7639	−0.601722
$321$	0	0
$322$	−13.0373	−0.726539
$323$	0	0
$324$	0	0
$325$	10.6861	0.592760
$326$	7.67396	0.425022
$327$	0	0
$328$	−28.4164	−1.56903
$329$	−10.7082	−0.590362
$330$	0	0
$331$	−24.2784	−1.33446	−0.667232	−	0.744850i	$-0.732521\pi$
$331$	−24.2784	−1.33446	−0.667232	+	0.744850i	$0.732521\pi$
$332$	0.763932	0.0419262
$333$	0	0
$334$	8.09017	0.442674
$335$	−0.212002	−0.0115829
$336$	0	0
$337$	−16.1150	−0.877838	−0.438919	−	0.898527i	$-0.644638\pi$
$337$	−16.1150	−0.877838	−0.438919	+	0.898527i	$0.644638\pi$
$338$	−4.14725	−0.225581
$339$	0	0
$340$	−1.88854	−0.102421
$341$	4.25325	0.230327
$342$	0	0
$343$	−16.7082	−0.902158
$344$	19.4702	1.04976
$345$	0	0
$346$	18.2918	0.983373
$347$	24.8328	1.33310	0.666548	−	0.745462i	$-0.267771\pi$
$347$	24.8328	1.33310	0.666548	+	0.745462i	$0.267771\pi$
$348$	0	0
$349$	11.3820	0.609263	0.304631	−	0.952470i	$-0.401467\pi$
$349$	11.3820	0.609263	0.304631	+	0.952470i	$0.401467\pi$
$350$	17.2905	0.924218
$351$	0	0
$352$	12.1392	0.647023
$353$	−27.6180	−1.46996	−0.734980	−	0.678089i	$-0.762808\pi$
$353$	−27.6180	−1.46996	−0.734980	+	0.678089i	$0.762808\pi$
$354$	0	0
$355$	−15.0049	−0.796378
$356$	10.9637	0.581072
$357$	0	0
$358$	−21.5066	−1.13666
$359$	37.6180	1.98540	0.992702	−	0.120594i	$-0.0384798\pi$
$359$	37.6180	1.98540	0.992702	+	0.120594i	$0.0384798\pi$
$360$	0	0
$361$	0	0
$362$	−28.9443	−1.52128
$363$	0	0
$364$	−8.05748	−0.422327
$365$	−11.1246	−0.582289
$366$	0	0
$367$	21.4721	1.12084	0.560418	−	0.828210i	$-0.310640\pi$
$367$	21.4721	1.12084	0.560418	+	0.828210i	$0.310640\pi$
$368$	−6.23607	−0.325078
$369$	0	0
$370$	−9.34752	−0.485955
$371$	−1.90211	−0.0987528
$372$	0	0
$373$	−4.31877	−0.223617	−0.111809	−	0.993730i	$-0.535664\pi$
$373$	−4.31877	−0.223617	−0.111809	+	0.993730i	$0.535664\pi$
$374$	−10.5146	−0.543698
$375$	0	0
$376$	−7.77997	−0.401221
$377$	8.09017	0.416665
$378$	0	0
$379$	−7.95148	−0.408440	−0.204220	−	0.978925i	$-0.565466\pi$
$379$	−7.95148	−0.408440	−0.204220	+	0.978925i	$0.565466\pi$
$380$	0	0
$381$	0	0
$382$	19.9847	1.02251
$383$	−14.0413	−0.717479	−0.358739	−	0.933438i	$-0.616793\pi$
$383$	−14.0413	−0.717479	−0.358739	+	0.933438i	$0.616793\pi$
$384$	0	0
$385$	18.9443	0.965489
$386$	−6.18034	−0.314571
$387$	0	0
$388$	−9.68208	−0.491533
$389$	−13.0902	−0.663698	−0.331849	−	0.943333i	$-0.607672\pi$
$389$	−13.0902	−0.663698	−0.331849	+	0.943333i	$0.607672\pi$
$390$	0	0
$391$	−6.47214	−0.327310
$392$	−33.6830	−1.70125
$393$	0	0
$394$	−2.97168	−0.149711
$395$	−9.40456	−0.473195
$396$	0	0
$397$	14.7082	0.738184	0.369092	−	0.929393i	$-0.379669\pi$
$397$	14.7082	0.738184	0.369092	+	0.929393i	$0.379669\pi$
$398$	−22.2703	−1.11631
$399$	0	0
$400$	8.27051	0.413525
$401$	7.60845	0.379948	0.189974	−	0.981789i	$-0.439160\pi$
$401$	7.60845	0.379948	0.189974	+	0.981789i	$0.439160\pi$
$402$	0	0
$403$	−3.61803	−0.180227
$404$	−8.61803	−0.428763
$405$	0	0
$406$	13.0902	0.649654
$407$	23.2744	1.15367
$408$	0	0
$409$	21.9273	1.08424	0.542118	−	0.840303i	$-0.317623\pi$
$409$	21.9273	1.08424	0.542118	+	0.840303i	$0.317623\pi$
$410$	−13.4164	−0.662589
$411$	0	0
$412$	0.620541	0.0305719
$413$	11.8617	0.583676
$414$	0	0
$415$	1.52786	0.0749999
$416$	−10.3262	−0.506285
$417$	0	0
$418$	0	0
$419$	33.4164	1.63250	0.816249	−	0.577700i	$-0.196050\pi$
$419$	33.4164	1.63250	0.816249	+	0.577700i	$0.196050\pi$
$420$	0	0
$421$	−14.1473	−0.689499	−0.344749	−	0.938695i	$-0.612036\pi$
$421$	−14.1473	−0.689499	−0.344749	+	0.938695i	$0.612036\pi$
$422$	−21.1803	−1.03104
$423$	0	0
$424$	−1.38197	−0.0671142
$425$	8.58359	0.416365
$426$	0	0
$427$	2.23607	0.108211
$428$	7.15942	0.346064
$429$	0	0
$430$	9.19256	0.443305
$431$	15.9434	0.767969	0.383984	−	0.923340i	$-0.374552\pi$
$431$	15.9434	0.767969	0.383984	+	0.923340i	$0.374552\pi$
$432$	0	0
$433$	−9.33905	−0.448806	−0.224403	−	0.974496i	$-0.572043\pi$
$433$	−9.33905	−0.448806	−0.224403	+	0.974496i	$0.572043\pi$
$434$	−5.85410	−0.281006
$435$	0	0
$436$	−0.449028	−0.0215045
$437$	0	0
$438$	0	0
$439$	−11.5842	−0.552883	−0.276442	−	0.961031i	$-0.589155\pi$
$439$	−11.5842	−0.552883	−0.276442	+	0.961031i	$0.589155\pi$
$440$	13.7638	0.656164
$441$	0	0
$442$	8.94427	0.425436
$443$	7.18034	0.341148	0.170574	−	0.985345i	$-0.445438\pi$
$443$	7.18034	0.341148	0.170574	+	0.985345i	$0.445438\pi$
$444$	0	0
$445$	21.9273	1.03945
$446$	15.9787	0.756614
$447$	0	0
$448$	−36.8885	−1.74282
$449$	34.9646	1.65008	0.825040	−	0.565074i	$-0.191152\pi$
$449$	34.9646	1.65008	0.825040	+	0.565074i	$0.191152\pi$
$450$	0	0
$451$	33.4055	1.57300
$452$	0	0
$453$	0	0
$454$	34.9230	1.63902
$455$	−16.1150	−0.755481
$456$	0	0
$457$	19.7082	0.921911	0.460955	−	0.887423i	$-0.347507\pi$
$457$	19.7082	0.921911	0.460955	+	0.887423i	$0.347507\pi$
$458$	4.80828	0.224676
$459$	0	0
$460$	2.00000	0.0932505
$461$	34.1803	1.59194	0.795969	−	0.605338i	$-0.206962\pi$
$461$	34.1803	1.59194	0.795969	+	0.605338i	$0.206962\pi$
$462$	0	0
$463$	16.8541	0.783277	0.391638	−	0.920119i	$-0.371908\pi$
$463$	16.8541	0.783277	0.391638	+	0.920119i	$0.371908\pi$
$464$	6.26137	0.290677
$465$	0	0
$466$	−15.8374	−0.733656
$467$	0.236068	0.0109239	0.00546196	−	0.999985i	$-0.498261\pi$
$467$	0.236068	0.0109239	0.00546196	+	0.999985i	$0.498261\pi$
$468$	0	0
$469$	−0.726543	−0.0335486
$470$	−3.67320	−0.169432
$471$	0	0
$472$	8.61803	0.396677
$473$	−22.8885	−1.05242
$474$	0	0
$475$	0	0
$476$	−6.47214	−0.296650
$477$	0	0
$478$	18.6376	0.852465
$479$	27.0902	1.23778	0.618891	−	0.785477i	$-0.287582\pi$
$479$	27.0902	1.23778	0.618891	+	0.785477i	$0.287582\pi$
$480$	0	0
$481$	−19.7984	−0.902728
$482$	26.6312	1.21302
$483$	0	0
$484$	−1.29180	−0.0587180
$485$	−19.3642	−0.879281
$486$	0	0
$487$	15.5599	0.705088	0.352544	−	0.935795i	$-0.385317\pi$
$487$	15.5599	0.705088	0.352544	+	0.935795i	$0.385317\pi$
$488$	1.62460	0.0735421
$489$	0	0
$490$	−15.9030	−0.718422
$491$	16.2705	0.734278	0.367139	−	0.930166i	$-0.380337\pi$
$491$	16.2705	0.734278	0.367139	+	0.930166i	$0.380337\pi$
$492$	0	0
$493$	6.49839	0.292673
$494$	0	0
$495$	0	0
$496$	−2.80017	−0.125731
$497$	−51.4226	−2.30662
$498$	0	0
$499$	21.5279	0.963720	0.481860	−	0.876248i	$-0.339961\pi$
$499$	21.5279	0.963720	0.481860	+	0.876248i	$0.339961\pi$
$500$	−6.47214	−0.289443
$501$	0	0
$502$	−12.2452	−0.546531
$503$	28.5967	1.27507	0.637533	−	0.770423i	$-0.279955\pi$
$503$	28.5967	1.27507	0.637533	+	0.770423i	$0.279955\pi$
$504$	0	0
$505$	−17.2361	−0.766995
$506$	11.1352	0.495018
$507$	0	0
$508$	9.85359	0.437182
$509$	−12.4167	−0.550362	−0.275181	−	0.961392i	$-0.588738\pi$
$509$	−12.4167	−0.550362	−0.275181	+	0.961392i	$0.588738\pi$
$510$	0	0
$511$	−38.1246	−1.68653
$512$	−22.6538	−1.00117
$513$	0	0
$514$	−10.6525	−0.469861
$515$	1.24108	0.0546886
$516$	0	0
$517$	9.14590	0.402236
$518$	−32.0344	−1.40751
$519$	0	0
$520$	−11.7082	−0.513439
$521$	−2.97168	−0.130192	−0.0650959	−	0.997879i	$-0.520735\pi$
$521$	−2.97168	−0.130192	−0.0650959	+	0.997879i	$0.520735\pi$
$522$	0	0
$523$	8.12299	0.355194	0.177597	−	0.984103i	$-0.443168\pi$
$523$	8.12299	0.355194	0.177597	+	0.984103i	$0.443168\pi$
$524$	−3.14590	−0.137429
$525$	0	0
$526$	−13.9758	−0.609375
$527$	−2.90617	−0.126595
$528$	0	0
$529$	−16.1459	−0.701996
$530$	−0.652476	−0.0283417
$531$	0	0
$532$	0	0
$533$	−28.4164	−1.23085
$534$	0	0
$535$	14.3188	0.619058
$536$	−0.527864	−0.0228003
$537$	0	0
$538$	9.06888	0.390987
$539$	39.5967	1.70555
$540$	0	0
$541$	−6.70820	−0.288408	−0.144204	−	0.989548i	$-0.546062\pi$
$541$	−6.70820	−0.288408	−0.144204	+	0.989548i	$0.546062\pi$
$542$	16.7760	0.720590
$543$	0	0
$544$	−8.29451	−0.355624
$545$	−0.898056	−0.0384685
$546$	0	0
$547$	29.2582	1.25099	0.625496	−	0.780227i	$-0.284897\pi$
$547$	29.2582	1.25099	0.625496	+	0.780227i	$0.284897\pi$
$548$	−2.90983	−0.124302
$549$	0	0
$550$	−14.7679	−0.629704
$551$	0	0
$552$	0	0
$553$	−32.2299	−1.37056
$554$	−22.6134	−0.960749
$555$	0	0
$556$	0.562306	0.0238471
$557$	−5.36068	−0.227139	−0.113570	−	0.993530i	$-0.536229\pi$
$557$	−5.36068	−0.227139	−0.113570	+	0.993530i	$0.536229\pi$
$558$	0	0
$559$	19.4702	0.823500
$560$	−12.4721	−0.527044
$561$	0	0
$562$	5.20163	0.219417
$563$	18.1231	0.763797	0.381898	−	0.924204i	$-0.375270\pi$
$563$	18.1231	0.763797	0.381898	+	0.924204i	$0.375270\pi$
$564$	0	0
$565$	0	0
$566$	−1.06957	−0.0449574
$567$	0	0
$568$	−37.3607	−1.56762
$569$	−18.3601	−0.769695	−0.384848	−	0.922980i	$-0.625746\pi$
$569$	−18.3601	−0.769695	−0.384848	+	0.922980i	$0.625746\pi$
$570$	0	0
$571$	41.1033	1.72012	0.860060	−	0.510192i	$-0.170426\pi$
$571$	41.1033	1.72012	0.860060	+	0.510192i	$0.170426\pi$
$572$	6.88191	0.287747
$573$	0	0
$574$	−45.9787	−1.91911
$575$	−9.09017	−0.379086
$576$	0	0
$577$	−28.7082	−1.19514	−0.597569	−	0.801817i	$-0.703867\pi$
$577$	−28.7082	−1.19514	−0.597569	+	0.801817i	$0.703867\pi$
$578$	−12.8003	−0.532420
$579$	0	0
$580$	−2.00811	−0.0833824
$581$	5.23607	0.217229
$582$	0	0
$583$	1.62460	0.0672840
$584$	−27.6992	−1.14620
$585$	0	0
$586$	−29.0689	−1.20082
$587$	−39.1803	−1.61715	−0.808573	−	0.588396i	$-0.799760\pi$
$587$	−39.1803	−1.61715	−0.808573	+	0.588396i	$0.799760\pi$
$588$	0	0
$589$	0	0
$590$	4.06888	0.167513
$591$	0	0
$592$	−15.3229	−0.629768
$593$	−30.7082	−1.26103	−0.630517	−	0.776175i	$-0.717157\pi$
$593$	−30.7082	−1.26103	−0.630517	+	0.776175i	$0.717157\pi$
$594$	0	0
$595$	−12.9443	−0.530663
$596$	−14.2361	−0.583132
$597$	0	0
$598$	−9.47214	−0.387344
$599$	20.6457	0.843562	0.421781	−	0.906698i	$-0.361405\pi$
$599$	20.6457	0.843562	0.421781	+	0.906698i	$0.361405\pi$
$600$	0	0
$601$	41.2915	1.68431	0.842157	−	0.539233i	$-0.181286\pi$
$601$	41.2915	1.68431	0.842157	+	0.539233i	$0.181286\pi$
$602$	31.5034	1.28398
$603$	0	0
$604$	−1.51860	−0.0617909
$605$	−2.58359	−0.105038
$606$	0	0
$607$	−1.51860	−0.0616380	−0.0308190	−	0.999525i	$-0.509812\pi$
$607$	−1.51860	−0.0616380	−0.0308190	+	0.999525i	$0.509812\pi$
$608$	0	0
$609$	0	0
$610$	0.767031	0.0310562
$611$	−7.77997	−0.314744
$612$	0	0
$613$	−24.5279	−0.990671	−0.495336	−	0.868702i	$-0.664955\pi$
$613$	−24.5279	−0.990671	−0.495336	+	0.868702i	$0.664955\pi$
$614$	−32.4377	−1.30908
$615$	0	0
$616$	47.1693	1.90051
$617$	−17.2705	−0.695285	−0.347642	−	0.937627i	$-0.613018\pi$
$617$	−17.2705	−0.695285	−0.347642	+	0.937627i	$0.613018\pi$
$618$	0	0
$619$	36.7771	1.47820	0.739098	−	0.673598i	$-0.235252\pi$
$619$	36.7771	1.47820	0.739098	+	0.673598i	$0.235252\pi$
$620$	0.898056	0.0360668
$621$	0	0
$622$	6.49839	0.260562
$623$	75.1460	3.01066
$624$	0	0
$625$	4.41641	0.176656
$626$	8.95554	0.357935
$627$	0	0
$628$	−9.61803	−0.383801
$629$	−15.9030	−0.634092
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	−23.4164	−0.931455
$633$	0	0
$634$	−10.4033	−0.413166
$635$	19.7072	0.782056
$636$	0	0
$637$	−33.6830	−1.33457
$638$	−11.1803	−0.442634
$639$	0	0
$640$	−4.35926	−0.172315
$641$	−28.9152	−1.14208	−0.571041	−	0.820921i	$-0.693460\pi$
$641$	−28.9152	−1.14208	−0.571041	+	0.820921i	$0.693460\pi$
$642$	0	0
$643$	7.70820	0.303982	0.151991	−	0.988382i	$-0.451432\pi$
$643$	7.70820	0.303982	0.151991	+	0.988382i	$0.451432\pi$
$644$	6.85410	0.270089
$645$	0	0
$646$	0	0
$647$	32.1246	1.26295	0.631474	−	0.775397i	$-0.282450\pi$
$647$	32.1246	1.26295	0.631474	+	0.775397i	$0.282450\pi$
$648$	0	0
$649$	−10.1311	−0.397681
$650$	12.5623	0.492734
$651$	0	0
$652$	−4.03444	−0.158001
$653$	−19.7426	−0.772589	−0.386295	−	0.922375i	$-0.626245\pi$
$653$	−19.7426	−0.772589	−0.386295	+	0.922375i	$0.626245\pi$
$654$	0	0
$655$	−6.29180	−0.245841
$656$	−21.9928	−0.858675
$657$	0	0
$658$	−12.5882	−0.490741
$659$	−18.6781	−0.727596	−0.363798	−	0.931478i	$-0.618520\pi$
$659$	−18.6781	−0.727596	−0.363798	+	0.931478i	$0.618520\pi$
$660$	0	0
$661$	5.04531	0.196240	0.0981199	−	0.995175i	$-0.468717\pi$
$661$	5.04531	0.196240	0.0981199	+	0.995175i	$0.468717\pi$
$662$	−28.5410	−1.10928
$663$	0	0
$664$	3.80423	0.147633
$665$	0	0
$666$	0	0
$667$	−6.88191	−0.266469
$668$	−4.25325	−0.164563
$669$	0	0
$670$	−0.249224	−0.00962835
$671$	−1.90983	−0.0737282
$672$	0	0
$673$	−28.2542	−1.08912	−0.544559	−	0.838722i	$-0.683303\pi$
$673$	−28.2542	−1.08912	−0.544559	+	0.838722i	$0.683303\pi$
$674$	−18.9443	−0.729706
$675$	0	0
$676$	2.18034	0.0838592
$677$	11.6902	0.449291	0.224645	−	0.974441i	$-0.427878\pi$
$677$	11.6902	0.449291	0.224645	+	0.974441i	$0.427878\pi$
$678$	0	0
$679$	−66.3620	−2.54674
$680$	−9.40456	−0.360649
$681$	0	0
$682$	5.00000	0.191460
$683$	−32.7445	−1.25293	−0.626466	−	0.779449i	$-0.715499\pi$
$683$	−32.7445	−1.25293	−0.626466	+	0.779449i	$0.715499\pi$
$684$	0	0
$685$	−5.81966	−0.222358
$686$	−19.6417	−0.749923
$687$	0	0
$688$	15.0689	0.574496
$689$	−1.38197	−0.0526487
$690$	0	0
$691$	46.3050	1.76152	0.880762	−	0.473560i	$-0.157031\pi$
$691$	46.3050	1.76152	0.880762	+	0.473560i	$0.157031\pi$
$692$	−9.61657	−0.365567
$693$	0	0
$694$	29.1927	1.10814
$695$	1.12461	0.0426590
$696$	0	0
$697$	−22.8254	−0.864572
$698$	13.3803	0.506452
$699$	0	0
$700$	−9.09017	−0.343576
$701$	33.7426	1.27444	0.637221	−	0.770681i	$-0.280084\pi$
$701$	33.7426	1.27444	0.637221	+	0.770681i	$0.280084\pi$
$702$	0	0
$703$	0	0
$704$	31.5066	1.18745
$705$	0	0
$706$	−32.4669	−1.22191
$707$	−59.0689	−2.22151
$708$	0	0
$709$	13.4164	0.503864	0.251932	−	0.967745i	$-0.418934\pi$
$709$	13.4164	0.503864	0.251932	+	0.967745i	$0.418934\pi$
$710$	−17.6393	−0.661992
$711$	0	0
$712$	54.5967	2.04610
$713$	3.07768	0.115260
$714$	0	0
$715$	13.7638	0.514738
$716$	11.3067	0.422550
$717$	0	0
$718$	44.2227	1.65037
$719$	−12.3820	−0.461769	−0.230885	−	0.972981i	$-0.574162\pi$
$719$	−12.3820	−0.461769	−0.230885	+	0.972981i	$0.574162\pi$
$720$	0	0
$721$	4.25325	0.158399
$722$	0	0
$723$	0	0
$724$	15.2169	0.565532
$725$	9.12705	0.338970
$726$	0	0
$727$	−19.0557	−0.706738	−0.353369	−	0.935484i	$-0.614964\pi$
$727$	−19.0557	−0.706738	−0.353369	+	0.935484i	$0.614964\pi$
$728$	−40.1246	−1.48712
$729$	0	0
$730$	−13.0778	−0.484030
$731$	15.6393	0.578441
$732$	0	0
$733$	35.1803	1.29942	0.649708	−	0.760184i	$-0.274891\pi$
$733$	35.1803	1.29942	0.649708	+	0.760184i	$0.274891\pi$
$734$	25.2420	0.931700
$735$	0	0
$736$	8.78402	0.323783
$737$	0.620541	0.0228579
$738$	0	0
$739$	10.5279	0.387274	0.193637	−	0.981073i	$-0.437972\pi$
$739$	10.5279	0.387274	0.193637	+	0.981073i	$0.437972\pi$
$740$	4.91428	0.180653
$741$	0	0
$742$	−2.23607	−0.0820886
$743$	−4.31877	−0.158440	−0.0792201	−	0.996857i	$-0.525243\pi$
$743$	−4.31877	−0.158440	−0.0792201	+	0.996857i	$0.525243\pi$
$744$	0	0
$745$	−28.4721	−1.04314
$746$	−5.07701	−0.185883
$747$	0	0
$748$	5.52786	0.202119
$749$	49.0714	1.79303
$750$	0	0
$751$	44.2631	1.61518	0.807592	−	0.589741i	$-0.200770\pi$
$751$	44.2631	1.61518	0.807592	+	0.589741i	$0.200770\pi$
$752$	−6.02129	−0.219574
$753$	0	0
$754$	9.51057	0.346354
$755$	−3.03719	−0.110535
$756$	0	0
$757$	−3.72949	−0.135551	−0.0677753	−	0.997701i	$-0.521590\pi$
$757$	−3.72949	−0.135551	−0.0677753	+	0.997701i	$0.521590\pi$
$758$	−9.34752	−0.339517
$759$	0	0
$760$	0	0
$761$	6.05573	0.219520	0.109760	−	0.993958i	$-0.464992\pi$
$761$	6.05573	0.219520	0.109760	+	0.993958i	$0.464992\pi$
$762$	0	0
$763$	−3.07768	−0.111420
$764$	−10.5066	−0.380115
$765$	0	0
$766$	−16.5066	−0.596407
$767$	8.61803	0.311179
$768$	0	0
$769$	−36.1591	−1.30393	−0.651964	−	0.758250i	$-0.726055\pi$
$769$	−36.1591	−1.30393	−0.651964	+	0.758250i	$0.726055\pi$
$770$	22.2703	0.802567
$771$	0	0
$772$	3.24920	0.116941
$773$	−37.0382	−1.33217	−0.666086	−	0.745875i	$-0.732032\pi$
$773$	−37.0382	−1.33217	−0.666086	+	0.745875i	$0.732032\pi$
$774$	0	0
$775$	−4.08174	−0.146620
$776$	−48.2148	−1.73081
$777$	0	0
$778$	−15.3884	−0.551702
$779$	0	0
$780$	0	0
$781$	43.9201	1.57159
$782$	−7.60845	−0.272078
$783$	0	0
$784$	−26.0689	−0.931032
$785$	−19.2361	−0.686565
$786$	0	0
$787$	14.3188	0.510412	0.255206	−	0.966887i	$-0.417857\pi$
$787$	14.3188	0.510412	0.255206	+	0.966887i	$0.417857\pi$
$788$	1.56231	0.0556548
$789$	0	0
$790$	−11.0557	−0.393345
$791$	0	0
$792$	0	0
$793$	1.62460	0.0576912
$794$	17.2905	0.613618
$795$	0	0
$796$	11.7082	0.414986
$797$	12.8658	0.455729	0.227864	−	0.973693i	$-0.426826\pi$
$797$	12.8658	0.455729	0.227864	+	0.973693i	$0.426826\pi$
$798$	0	0
$799$	−6.24922	−0.221082
$800$	−11.6497	−0.411879
$801$	0	0
$802$	8.94427	0.315833
$803$	32.5623	1.14910
$804$	0	0
$805$	13.7082	0.483151
$806$	−4.25325	−0.149814
$807$	0	0
$808$	−42.9161	−1.50978
$809$	0.729490	0.0256475	0.0128238	−	0.999918i	$-0.495918\pi$
$809$	0.729490	0.0256475	0.0128238	+	0.999918i	$0.495918\pi$
$810$	0	0
$811$	32.1239	1.12802	0.564012	−	0.825767i	$-0.309257\pi$
$811$	32.1239	1.12802	0.564012	+	0.825767i	$0.309257\pi$
$812$	−6.88191	−0.241508
$813$	0	0
$814$	27.3607	0.958991
$815$	−8.06888	−0.282641
$816$	0	0
$817$	0	0
$818$	25.7771	0.901275
$819$	0	0
$820$	7.05342	0.246316
$821$	19.0344	0.664307	0.332153	−	0.943225i	$-0.392225\pi$
$821$	19.0344	0.664307	0.332153	+	0.943225i	$0.392225\pi$
$822$	0	0
$823$	−38.7214	−1.34974	−0.674871	−	0.737936i	$-0.735801\pi$
$823$	−38.7214	−1.34974	−0.674871	+	0.737936i	$0.735801\pi$
$824$	3.09017	0.107651
$825$	0	0
$826$	13.9443	0.485183
$827$	15.7314	0.547036	0.273518	−	0.961867i	$-0.411813\pi$
$827$	15.7314	0.547036	0.273518	+	0.961867i	$0.411813\pi$
$828$	0	0
$829$	−11.6902	−0.406017	−0.203009	−	0.979177i	$-0.565072\pi$
$829$	−11.6902	−0.406017	−0.203009	+	0.979177i	$0.565072\pi$
$830$	1.79611	0.0623440
$831$	0	0
$832$	−26.8011	−0.929161
$833$	−27.0557	−0.937425
$834$	0	0
$835$	−8.50651	−0.294380
$836$	0	0
$837$	0	0
$838$	39.2833	1.35702
$839$	23.6174	0.815364	0.407682	−	0.913124i	$-0.366337\pi$
$839$	23.6174	0.815364	0.407682	+	0.913124i	$0.366337\pi$
$840$	0	0
$841$	−22.0902	−0.761730
$842$	−16.6312	−0.573149
$843$	0	0
$844$	11.1352	0.383288
$845$	4.36068	0.150012
$846$	0	0
$847$	−8.85410	−0.304231
$848$	−1.06957	−0.0367292
$849$	0	0
$850$	10.0906	0.346105
$851$	16.8415	0.577319
$852$	0	0
$853$	−22.3050	−0.763707	−0.381853	−	0.924223i	$-0.624714\pi$
$853$	−22.3050	−0.763707	−0.381853	+	0.924223i	$0.624714\pi$
$854$	2.62866	0.0899507
$855$	0	0
$856$	35.6525	1.21858
$857$	21.0543	0.719200	0.359600	−	0.933107i	$-0.382913\pi$
$857$	21.0543	0.719200	0.359600	+	0.933107i	$0.382913\pi$
$858$	0	0
$859$	21.0557	0.718412	0.359206	−	0.933258i	$-0.383048\pi$
$859$	21.0557	0.718412	0.359206	+	0.933258i	$0.383048\pi$
$860$	−4.83282	−0.164798
$861$	0	0
$862$	18.7426	0.638377
$863$	−27.1846	−0.925375	−0.462687	−	0.886521i	$-0.653115\pi$
$863$	−27.1846	−0.925375	−0.462687	+	0.886521i	$0.653115\pi$
$864$	0	0
$865$	−19.2331	−0.653946
$866$	−10.9787	−0.373072
$867$	0	0
$868$	3.07768	0.104463
$869$	27.5276	0.933811
$870$	0	0
$871$	−0.527864	−0.0178860
$872$	−2.23607	−0.0757228
$873$	0	0
$874$	0	0
$875$	−44.3607	−1.49966
$876$	0	0
$877$	41.2915	1.39431	0.697157	−	0.716919i	$-0.254448\pi$
$877$	41.2915	1.39431	0.697157	+	0.716919i	$0.254448\pi$
$878$	−13.6180	−0.459586
$879$	0	0
$880$	10.6525	0.359095
$881$	55.5755	1.87238	0.936192	−	0.351488i	$-0.114324\pi$
$881$	55.5755	1.87238	0.936192	+	0.351488i	$0.114324\pi$
$882$	0	0
$883$	43.0344	1.44822	0.724112	−	0.689682i	$-0.242250\pi$
$883$	43.0344	1.44822	0.724112	+	0.689682i	$0.242250\pi$
$884$	−4.70228	−0.158155
$885$	0	0
$886$	8.44100	0.283581
$887$	−26.6296	−0.894134	−0.447067	−	0.894501i	$-0.647531\pi$
$887$	−26.6296	−0.894134	−0.447067	+	0.894501i	$0.647531\pi$
$888$	0	0
$889$	67.5375	2.26514
$890$	25.7771	0.864050
$891$	0	0
$892$	−8.40051	−0.281270
$893$	0	0
$894$	0	0
$895$	22.6134	0.755881
$896$	−14.9394	−0.499090
$897$	0	0
$898$	41.1033	1.37164
$899$	−3.09017	−0.103063
$900$	0	0
$901$	−1.11006	−0.0369814
$902$	39.2705	1.30756
$903$	0	0
$904$	0	0
$905$	30.4338	1.01165
$906$	0	0
$907$	12.6538	0.420161	0.210081	−	0.977684i	$-0.432627\pi$
$907$	12.6538	0.420161	0.210081	+	0.977684i	$0.432627\pi$
$908$	−18.3601	−0.609301
$909$	0	0
$910$	−18.9443	−0.627996
$911$	−18.3601	−0.608297	−0.304149	−	0.952625i	$-0.598372\pi$
$911$	−18.3601	−0.608297	−0.304149	+	0.952625i	$0.598372\pi$
$912$	0	0
$913$	−4.47214	−0.148006
$914$	23.1684	0.766342
$915$	0	0
$916$	−2.52786	−0.0835230
$917$	−21.5623	−0.712050
$918$	0	0
$919$	−2.63932	−0.0870631	−0.0435316	−	0.999052i	$-0.513861\pi$
$919$	−2.63932	−0.0870631	−0.0435316	+	0.999052i	$0.513861\pi$
$920$	9.95959	0.328358
$921$	0	0
$922$	40.1814	1.32330
$923$	−37.3607	−1.22974
$924$	0	0
$925$	−22.3358	−0.734398
$926$	19.8132	0.651102
$927$	0	0
$928$	−8.81966	−0.289520
$929$	21.5066	0.705608	0.352804	−	0.935697i	$-0.385228\pi$
$929$	21.5066	0.705608	0.352804	+	0.935697i	$0.385228\pi$
$930$	0	0
$931$	0	0
$932$	8.32624	0.272735
$933$	0	0
$934$	0.277515	0.00908055
$935$	11.0557	0.361561
$936$	0	0
$937$	−41.7984	−1.36549	−0.682747	−	0.730655i	$-0.739215\pi$
$937$	−41.7984	−1.36549	−0.682747	+	0.730655i	$0.739215\pi$
$938$	−0.854102	−0.0278874
$939$	0	0
$940$	1.93112	0.0629861
$941$	−14.5559	−0.474508	−0.237254	−	0.971448i	$-0.576247\pi$
$941$	−14.5559	−0.474508	−0.237254	+	0.971448i	$0.576247\pi$
$942$	0	0
$943$	24.1724	0.787163
$944$	6.66991	0.217087
$945$	0	0
$946$	−26.9071	−0.874825
$947$	5.81966	0.189114	0.0945568	−	0.995519i	$-0.469857\pi$
$947$	5.81966	0.189114	0.0945568	+	0.995519i	$0.469857\pi$
$948$	0	0
$949$	−27.6992	−0.899153
$950$	0	0
$951$	0	0
$952$	−32.2299	−1.04458
$953$	−37.3157	−1.20877	−0.604387	−	0.796691i	$-0.706582\pi$
$953$	−37.3157	−1.20877	−0.604387	+	0.796691i	$0.706582\pi$
$954$	0	0
$955$	−21.0132	−0.679970
$956$	−9.79837	−0.316902
$957$	0	0
$958$	31.8464	1.02891
$959$	−19.9443	−0.644034
$960$	0	0
$961$	−29.6180	−0.955420
$962$	−23.2744	−0.750396
$963$	0	0
$964$	−14.0008	−0.450937
$965$	6.49839	0.209191
$966$	0	0
$967$	33.7082	1.08398	0.541991	−	0.840384i	$-0.317671\pi$
$967$	33.7082	1.08398	0.541991	+	0.840384i	$0.317671\pi$
$968$	−6.43288	−0.206761
$969$	0	0
$970$	−22.7639	−0.730906
$971$	17.0785	0.548076	0.274038	−	0.961719i	$-0.411641\pi$
$971$	17.0785	0.548076	0.274038	+	0.961719i	$0.411641\pi$
$972$	0	0
$973$	3.85410	0.123557
$974$	18.2918	0.586107
$975$	0	0
$976$	1.25735	0.0402469
$977$	28.2542	0.903931	0.451966	−	0.892035i	$-0.350723\pi$
$977$	28.2542	0.903931	0.451966	+	0.892035i	$0.350723\pi$
$978$	0	0
$979$	−64.1823	−2.05128
$980$	8.36068	0.267072
$981$	0	0
$982$	19.1271	0.610371
$983$	21.6093	0.689230	0.344615	−	0.938744i	$-0.388010\pi$
$983$	21.6093	0.689230	0.344615	+	0.938744i	$0.388010\pi$
$984$	0	0
$985$	3.12461	0.0995584
$986$	7.63932	0.243286
$987$	0	0
$988$	0	0
$989$	−16.5623	−0.526651
$990$	0	0
$991$	21.0543	0.668811	0.334405	−	0.942429i	$-0.391465\pi$
$991$	21.0543	0.668811	0.334405	+	0.942429i	$0.391465\pi$
$992$	3.94427	0.125231
$993$	0	0
$994$	−60.4508	−1.91738
$995$	23.4164	0.742350
$996$	0	0
$997$	7.12461	0.225639	0.112819	−	0.993616i	$-0.464012\pi$
$997$	7.12461	0.225639	0.112819	+	0.993616i	$0.464012\pi$
$998$	25.3075	0.801096
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.bc.1.3		4
3.2	odd	2		361.2.a.i.1.2	✓	4
12.11	even	2		5776.2.a.bu.1.2		4
15.14	odd	2		9025.2.a.bj.1.3		4
19.18	odd	2	inner	3249.2.a.bc.1.2		4
57.2	even	18		361.2.e.m.99.2		24
57.5	odd	18		361.2.e.m.234.2		24
57.8	even	6		361.2.c.j.292.2		8
57.11	odd	6		361.2.c.j.292.3		8
57.14	even	18		361.2.e.m.234.3		24
57.17	odd	18		361.2.e.m.99.3		24
57.23	odd	18		361.2.e.m.54.2		24
57.26	odd	6		361.2.c.j.68.3		8
57.29	even	18		361.2.e.m.62.2		24
57.32	even	18		361.2.e.m.245.3		24
57.35	odd	18		361.2.e.m.28.2		24
57.41	even	18		361.2.e.m.28.3		24
57.44	odd	18		361.2.e.m.245.2		24
57.47	odd	18		361.2.e.m.62.3		24
57.50	even	6		361.2.c.j.68.2		8
57.53	even	18		361.2.e.m.54.3		24
57.56	even	2		361.2.a.i.1.3	yes	4
228.227	odd	2		5776.2.a.bu.1.3		4
285.284	even	2		9025.2.a.bj.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
361.2.a.i.1.2	✓	4	3.2	odd	2
361.2.a.i.1.3	yes	4	57.56	even	2
361.2.c.j.68.2		8	57.50	even	6
361.2.c.j.68.3		8	57.26	odd	6
361.2.c.j.292.2		8	57.8	even	6
361.2.c.j.292.3		8	57.11	odd	6
361.2.e.m.28.2		24	57.35	odd	18
361.2.e.m.28.3		24	57.41	even	18
361.2.e.m.54.2		24	57.23	odd	18
361.2.e.m.54.3		24	57.53	even	18
361.2.e.m.62.2		24	57.29	even	18
361.2.e.m.62.3		24	57.47	odd	18
361.2.e.m.99.2		24	57.2	even	18
361.2.e.m.99.3		24	57.17	odd	18
361.2.e.m.234.2		24	57.5	odd	18
361.2.e.m.234.3		24	57.14	even	18
361.2.e.m.245.2		24	57.44	odd	18
361.2.e.m.245.3		24	57.32	even	18
3249.2.a.bc.1.2		4	19.18	odd	2	inner
3249.2.a.bc.1.3		4	1.1	even	1	trivial
5776.2.a.bu.1.2		4	12.11	even	2
5776.2.a.bu.1.3		4	228.227	odd	2
9025.2.a.bj.1.2		4	285.284	even	2
9025.2.a.bj.1.3		4	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17557 q^{2} -0.618034 q^{4} -1.23607 q^{5} -4.23607 q^{7} -3.07768 q^{8} -1.45309 q^{10} +3.61803 q^{11} -3.07768 q^{13} -4.97980 q^{14} -2.38197 q^{16} -2.47214 q^{17} +0.763932 q^{20} +4.25325 q^{22} +2.61803 q^{23} -3.47214 q^{25} -3.61803 q^{26} +2.61803 q^{28} -2.62866 q^{29} +1.17557 q^{31} +3.35520 q^{32} -2.90617 q^{34} +5.23607 q^{35} +6.43288 q^{37} +3.80423 q^{40} +9.23305 q^{41} -6.32624 q^{43} -2.23607 q^{44} +3.07768 q^{46} +2.52786 q^{47} +10.9443 q^{49} -4.08174 q^{50} +1.90211 q^{52} +0.449028 q^{53} -4.47214 q^{55} +13.0373 q^{56} -3.09017 q^{58} -2.80017 q^{59} -0.527864 q^{61} +1.38197 q^{62} +8.70820 q^{64} +3.80423 q^{65} +0.171513 q^{67} +1.52786 q^{68} +6.15537 q^{70} +12.1392 q^{71} +9.00000 q^{73} +7.56231 q^{74} -15.3262 q^{77} +7.60845 q^{79} +2.94427 q^{80} +10.8541 q^{82} -1.23607 q^{83} +3.05573 q^{85} -7.43694 q^{86} -11.1352 q^{88} -17.7396 q^{89} +13.0373 q^{91} -1.61803 q^{92} +2.97168 q^{94} +15.6659 q^{97} +12.8658 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 4 q^{5} - 8 q^{7} + 10 q^{11} - 14 q^{16} + 8 q^{17} + 12 q^{20} + 6 q^{23} + 4 q^{25} - 10 q^{26} + 6 q^{28} + 12 q^{35} + 6 q^{43} + 28 q^{47} + 8 q^{49} + 10 q^{58} - 20 q^{61} + 10 q^{62}+ \cdots - 2 q^{92}+O(q^{100}) \) 4 * q + 2 * q^4 + 4 * q^5 - 8 * q^7 + 10 * q^11 - 14 * q^16 + 8 * q^17 + 12 * q^20 + 6 * q^23 + 4 * q^25 - 10 * q^26 + 6 * q^28 + 12 * q^35 + 6 * q^43 + 28 * q^47 + 8 * q^49 + 10 * q^58 - 20 * q^61 + 10 * q^62 + 8 * q^64 + 24 * q^68 + 36 * q^73 - 10 * q^74 - 30 * q^77 - 24 * q^80 + 30 * q^82 + 4 * q^83 + 48 * q^85 - 2 * q^92

Introduction

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