LMFDB - Embedded newform 1449.4.a.m.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,4,Mod(1,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1449.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1449.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:	$85.4937675983$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 57x^{7} - 13x^{6} + 1042x^{5} + 331x^{4} - 6570x^{3} - 1782x^{2} + 9424x + 5112 $ x^9 - 57x^7 - 13x^6 + 1042x^5 + 331x^4 - 6570x^3 - 1782x^2 + 9424*x + 5112
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.05204$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.05204 q^{2} -6.89321 q^{4} +11.5074 q^{5} +7.00000 q^{7} +15.6682 q^{8} +O(q^{10})$	\(q-1.05204 q^{2} -6.89321 q^{4} +11.5074 q^{5} +7.00000 q^{7} +15.6682 q^{8} -12.1063 q^{10} +58.4398 q^{11} -37.9629 q^{13} -7.36428 q^{14} +38.6621 q^{16} +131.447 q^{17} +60.3971 q^{19} -79.3232 q^{20} -61.4809 q^{22} -23.0000 q^{23} +7.42099 q^{25} +39.9384 q^{26} -48.2525 q^{28} +124.917 q^{29} +294.792 q^{31} -166.020 q^{32} -138.287 q^{34} +80.5520 q^{35} -43.6444 q^{37} -63.5401 q^{38} +180.301 q^{40} +174.663 q^{41} -56.1216 q^{43} -402.838 q^{44} +24.1969 q^{46} -256.458 q^{47} +49.0000 q^{49} -7.80717 q^{50} +261.686 q^{52} -525.286 q^{53} +672.492 q^{55} +109.678 q^{56} -131.417 q^{58} -35.2060 q^{59} +456.577 q^{61} -310.133 q^{62} -134.637 q^{64} -436.855 q^{65} +610.737 q^{67} -906.091 q^{68} -84.7439 q^{70} +397.663 q^{71} -450.519 q^{73} +45.9156 q^{74} -416.330 q^{76} +409.078 q^{77} -891.829 q^{79} +444.901 q^{80} -183.753 q^{82} -1306.14 q^{83} +1512.62 q^{85} +59.0421 q^{86} +915.649 q^{88} +125.040 q^{89} -265.740 q^{91} +158.544 q^{92} +269.804 q^{94} +695.015 q^{95} -179.753 q^{97} -51.5499 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 42 q^{4} - 29 q^{5} + 63 q^{7} + 39 q^{8}+O(q^{10}) $ 9 * q + 42 * q^4 - 29 * q^5 + 63 * q^7 + 39 * q^8	$ 9 q + 42 q^{4} - 29 q^{5} + 63 q^{7} + 39 q^{8} - 55 q^{10} - 12 q^{11} + 199 q^{13} + 170 q^{16} - 116 q^{17} + 260 q^{19} - 324 q^{20} - 265 q^{22} - 207 q^{23} + 438 q^{25} + 270 q^{26} + 294 q^{28} + 107 q^{29} + 440 q^{31} + 802 q^{32} + 295 q^{34} - 203 q^{35} + 563 q^{37} + 569 q^{38} - 640 q^{40} - 243 q^{41} + 435 q^{43} - 1025 q^{44} + 133 q^{47} + 441 q^{49} - 104 q^{50} + 2693 q^{52} - 958 q^{53} + 1846 q^{55} + 273 q^{56} + 2796 q^{58} - 538 q^{59} + 1374 q^{61} - 1263 q^{62} - 83 q^{64} - 745 q^{65} + 752 q^{67} - 5593 q^{68} - 385 q^{70} + 418 q^{71} + 2406 q^{73} - 352 q^{74} + 2765 q^{76} - 84 q^{77} - 486 q^{79} - 5709 q^{80} + 2726 q^{82} - 106 q^{83} + 4130 q^{85} + 2576 q^{86} + 1270 q^{88} - 234 q^{89} + 1393 q^{91} - 966 q^{92} + 4967 q^{94} + 3074 q^{95} + 2409 q^{97}+O(q^{100}) $ 9 * q + 42 * q^4 - 29 * q^5 + 63 * q^7 + 39 * q^8 - 55 * q^10 - 12 * q^11 + 199 * q^13 + 170 * q^16 - 116 * q^17 + 260 * q^19 - 324 * q^20 - 265 * q^22 - 207 * q^23 + 438 * q^25 + 270 * q^26 + 294 * q^28 + 107 * q^29 + 440 * q^31 + 802 * q^32 + 295 * q^34 - 203 * q^35 + 563 * q^37 + 569 * q^38 - 640 * q^40 - 243 * q^41 + 435 * q^43 - 1025 * q^44 + 133 * q^47 + 441 * q^49 - 104 * q^50 + 2693 * q^52 - 958 * q^53 + 1846 * q^55 + 273 * q^56 + 2796 * q^58 - 538 * q^59 + 1374 * q^61 - 1263 * q^62 - 83 * q^64 - 745 * q^65 + 752 * q^67 - 5593 * q^68 - 385 * q^70 + 418 * q^71 + 2406 * q^73 - 352 * q^74 + 2765 * q^76 - 84 * q^77 - 486 * q^79 - 5709 * q^80 + 2726 * q^82 - 106 * q^83 + 4130 * q^85 + 2576 * q^86 + 1270 * q^88 - 234 * q^89 + 1393 * q^91 - 966 * q^92 + 4967 * q^94 + 3074 * q^95 + 2409 * 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 $

$2$	−1.05204	−0.371952	−0.185976	−	0.982554i	$-0.559545\pi$
$2$	−1.05204	−0.371952	−0.185976	+	0.982554i	$0.559545\pi$
$3$	0	0
$3$	0	0
$4$	−6.89321	−0.861652
$5$	11.5074	1.02926	0.514628	−	0.857414i	$-0.327930\pi$
$5$	11.5074	1.02926	0.514628	+	0.857414i	$0.327930\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	15.6682	0.692445
$9$	0	0
$10$	−12.1063	−0.382834
$11$	58.4398	1.60184	0.800920	−	0.598771i	$-0.204344\pi$
$11$	58.4398	1.60184	0.800920	+	0.598771i	$0.204344\pi$
$12$	0	0
$13$	−37.9629	−0.809923	−0.404962	−	0.914334i	$-0.632715\pi$
$13$	−37.9629	−0.809923	−0.404962	+	0.914334i	$0.632715\pi$
$14$	−7.36428	−0.140585
$15$	0	0
$16$	38.6621	0.604095
$17$	131.447	1.87533	0.937663	−	0.347545i	$-0.112985\pi$
$17$	131.447	1.87533	0.937663	+	0.347545i	$0.112985\pi$
$18$	0	0
$19$	60.3971	0.729265	0.364633	−	0.931151i	$-0.381195\pi$
$19$	60.3971	0.729265	0.364633	+	0.931151i	$0.381195\pi$
$20$	−79.3232	−0.886860
$21$	0	0
$22$	−61.4809	−0.595808
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	7.42099	0.0593679
$26$	39.9384	0.301253
$27$	0	0
$28$	−48.2525	−0.325674
$29$	124.917	0.799876	0.399938	−	0.916542i	$-0.369032\pi$
$29$	124.917	0.799876	0.399938	+	0.916542i	$0.369032\pi$
$30$	0	0
$31$	294.792	1.70794	0.853971	−	0.520321i	$-0.174188\pi$
$31$	294.792	1.70794	0.853971	+	0.520321i	$0.174188\pi$
$32$	−166.020	−0.917140
$33$	0	0
$34$	−138.287	−0.697532
$35$	80.5520	0.389022
$36$	0	0
$37$	−43.6444	−0.193921	−0.0969607	−	0.995288i	$-0.530912\pi$
$37$	−43.6444	−0.193921	−0.0969607	+	0.995288i	$0.530912\pi$
$38$	−63.5401	−0.271252
$39$	0	0
$40$	180.301	0.712703
$41$	174.663	0.665313	0.332656	−	0.943048i	$-0.392055\pi$
$41$	174.663	0.665313	0.332656	+	0.943048i	$0.392055\pi$
$42$	0	0
$43$	−56.1216	−0.199034	−0.0995170	−	0.995036i	$-0.531730\pi$
$43$	−56.1216	−0.199034	−0.0995170	+	0.995036i	$0.531730\pi$
$44$	−402.838	−1.38023
$45$	0	0
$46$	24.1969	0.0775574
$47$	−256.458	−0.795921	−0.397961	−	0.917402i	$-0.630282\pi$
$47$	−256.458	−0.795921	−0.397961	+	0.917402i	$0.630282\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−7.80717	−0.0220820
$51$	0	0
$52$	261.686	0.697872
$53$	−525.286	−1.36139	−0.680695	−	0.732567i	$-0.738322\pi$
$53$	−525.286	−1.36139	−0.680695	+	0.732567i	$0.738322\pi$
$54$	0	0
$55$	672.492	1.64870
$56$	109.678	0.261720
$57$	0	0
$58$	−131.417	−0.297516
$59$	−35.2060	−0.0776852	−0.0388426	−	0.999245i	$-0.512367\pi$
$59$	−35.2060	−0.0776852	−0.0388426	+	0.999245i	$0.512367\pi$
$60$	0	0
$61$	456.577	0.958339	0.479169	−	0.877722i	$-0.340938\pi$
$61$	456.577	0.958339	0.479169	+	0.877722i	$0.340938\pi$
$62$	−310.133	−0.635272
$63$	0	0
$64$	−134.637	−0.262963
$65$	−436.855	−0.833619
$66$	0	0
$67$	610.737	1.11363	0.556817	−	0.830635i	$-0.312023\pi$
$67$	610.737	1.11363	0.556817	+	0.830635i	$0.312023\pi$
$68$	−906.091	−1.61588
$69$	0	0
$70$	−84.7439	−0.144698
$71$	397.663	0.664702	0.332351	−	0.943156i	$-0.392158\pi$
$71$	397.663	0.664702	0.332351	+	0.943156i	$0.392158\pi$
$72$	0	0
$73$	−450.519	−0.722318	−0.361159	−	0.932504i	$-0.617619\pi$
$73$	−450.519	−0.722318	−0.361159	+	0.932504i	$0.617619\pi$
$74$	45.9156	0.0721295
$75$	0	0
$76$	−416.330	−0.628372
$77$	409.078	0.605439
$78$	0	0
$79$	−891.829	−1.27011	−0.635054	−	0.772468i	$-0.719022\pi$
$79$	−891.829	−1.27011	−0.635054	+	0.772468i	$0.719022\pi$
$80$	444.901	0.621769
$81$	0	0
$82$	−183.753	−0.247464
$83$	−1306.14	−1.72732	−0.863658	−	0.504077i	$-0.831833\pi$
$83$	−1306.14	−1.72732	−0.863658	+	0.504077i	$0.831833\pi$
$84$	0	0
$85$	1512.62	1.93019
$86$	59.0421	0.0740311
$87$	0	0
$88$	915.649	1.10919
$89$	125.040	0.148924	0.0744619	−	0.997224i	$-0.476276\pi$
$89$	125.040	0.148924	0.0744619	+	0.997224i	$0.476276\pi$
$90$	0	0
$91$	−265.740	−0.306122
$92$	158.544	0.179667
$93$	0	0
$94$	269.804	0.296045
$95$	695.015	0.750600
$96$	0	0
$97$	−179.753	−0.188157	−0.0940783	−	0.995565i	$-0.529990\pi$
$97$	−179.753	−0.188157	−0.0940783	+	0.995565i	$0.529990\pi$
$98$	−51.5499	−0.0531360
$99$	0	0
$100$	−51.1545	−0.0511545
$101$	1897.29	1.86919	0.934593	−	0.355718i	$-0.115764\pi$
$101$	1897.29	1.86919	0.934593	+	0.355718i	$0.115764\pi$
$102$	0	0
$103$	−1402.83	−1.34199	−0.670993	−	0.741463i	$-0.734132\pi$
$103$	−1402.83	−1.34199	−0.670993	+	0.741463i	$0.734132\pi$
$104$	−594.812	−0.560828
$105$	0	0
$106$	552.622	0.506372
$107$	−671.354	−0.606563	−0.303281	−	0.952901i	$-0.598082\pi$
$107$	−671.354	−0.606563	−0.303281	+	0.952901i	$0.598082\pi$
$108$	0	0
$109$	−685.095	−0.602020	−0.301010	−	0.953621i	$-0.597324\pi$
$109$	−685.095	−0.602020	−0.301010	+	0.953621i	$0.597324\pi$
$110$	−707.488	−0.613239
$111$	0	0
$112$	270.635	0.228327
$113$	742.519	0.618144	0.309072	−	0.951039i	$-0.399982\pi$
$113$	742.519	0.618144	0.309072	+	0.951039i	$0.399982\pi$
$114$	0	0
$115$	−264.671	−0.214615
$116$	−861.076	−0.689215
$117$	0	0
$118$	37.0381	0.0288952
$119$	920.128	0.708807
$120$	0	0
$121$	2084.21	1.56589
$122$	−480.337	−0.356456
$123$	0	0
$124$	−2032.06	−1.47165
$125$	−1353.03	−0.968151
$126$	0	0
$127$	−2207.73	−1.54256	−0.771279	−	0.636498i	$-0.780383\pi$
$127$	−2207.73	−1.54256	−0.771279	+	0.636498i	$0.780383\pi$
$128$	1469.80	1.01495
$129$	0	0
$130$	459.589	0.310066
$131$	−954.191	−0.636398	−0.318199	−	0.948024i	$-0.603078\pi$
$131$	−954.191	−0.636398	−0.318199	+	0.948024i	$0.603078\pi$
$132$	0	0
$133$	422.779	0.275636
$134$	−642.520	−0.414218
$135$	0	0
$136$	2059.54	1.29856
$137$	−944.268	−0.588863	−0.294432	−	0.955673i	$-0.595130\pi$
$137$	−944.268	−0.588863	−0.294432	+	0.955673i	$0.595130\pi$
$138$	0	0
$139$	2724.16	1.66230	0.831152	−	0.556045i	$-0.187682\pi$
$139$	2724.16	1.66230	0.831152	+	0.556045i	$0.187682\pi$
$140$	−555.262	−0.335202
$141$	0	0
$142$	−418.357	−0.247237
$143$	−2218.54	−1.29737
$144$	0	0
$145$	1437.47	0.823278
$146$	473.963	0.268668
$147$	0	0
$148$	300.850	0.167093
$149$	2103.44	1.15651	0.578257	−	0.815855i	$-0.303733\pi$
$149$	2103.44	1.15651	0.578257	+	0.815855i	$0.303733\pi$
$150$	0	0
$151$	100.617	0.0542256	0.0271128	−	0.999632i	$-0.491369\pi$
$151$	100.617	0.0542256	0.0271128	+	0.999632i	$0.491369\pi$
$152$	946.316	0.504976
$153$	0	0
$154$	−430.367	−0.225194
$155$	3392.30	1.75791
$156$	0	0
$157$	1097.53	0.557912	0.278956	−	0.960304i	$-0.410012\pi$
$157$	1097.53	0.557912	0.278956	+	0.960304i	$0.410012\pi$
$158$	938.239	0.472420
$159$	0	0
$160$	−1910.46	−0.943972
$161$	−161.000	−0.0788110
$162$	0	0
$163$	795.478	0.382249	0.191125	−	0.981566i	$-0.438787\pi$
$163$	795.478	0.382249	0.191125	+	0.981566i	$0.438787\pi$
$164$	−1203.99	−0.573268
$165$	0	0
$166$	1374.11	0.642479
$167$	3693.50	1.71145	0.855724	−	0.517433i	$-0.173112\pi$
$167$	3693.50	1.71145	0.855724	+	0.517433i	$0.173112\pi$
$168$	0	0
$169$	−755.821	−0.344024
$170$	−1591.33	−0.717939
$171$	0	0
$172$	386.858	0.171498
$173$	−2127.03	−0.934771	−0.467386	−	0.884054i	$-0.654804\pi$
$173$	−2127.03	−0.934771	−0.467386	+	0.884054i	$0.654804\pi$
$174$	0	0
$175$	51.9469	0.0224390
$176$	2259.40	0.967664
$177$	0	0
$178$	−131.547	−0.0553925
$179$	1724.06	0.719899	0.359950	−	0.932972i	$-0.382794\pi$
$179$	1724.06	0.719899	0.359950	+	0.932972i	$0.382794\pi$
$180$	0	0
$181$	2490.59	1.02279	0.511393	−	0.859347i	$-0.329130\pi$
$181$	2490.59	1.02279	0.511393	+	0.859347i	$0.329130\pi$
$182$	279.569	0.113863
$183$	0	0
$184$	−360.370	−0.144385
$185$	−502.235	−0.199595
$186$	0	0
$187$	7681.72	3.00397
$188$	1767.82	0.685807
$189$	0	0
$190$	−731.183	−0.279187
$191$	2876.47	1.08971	0.544853	−	0.838531i	$-0.316585\pi$
$191$	2876.47	1.08971	0.544853	+	0.838531i	$0.316585\pi$
$192$	0	0
$193$	−1680.29	−0.626685	−0.313342	−	0.949640i	$-0.601449\pi$
$193$	−1680.29	−0.626685	−0.313342	+	0.949640i	$0.601449\pi$
$194$	189.108	0.0699852
$195$	0	0
$196$	−337.767	−0.123093
$197$	−4315.09	−1.56060	−0.780298	−	0.625408i	$-0.784932\pi$
$197$	−4315.09	−1.56060	−0.780298	+	0.625408i	$0.784932\pi$
$198$	0	0
$199$	2705.03	0.963589	0.481794	−	0.876284i	$-0.339985\pi$
$199$	2705.03	0.963589	0.481794	+	0.876284i	$0.339985\pi$
$200$	116.274	0.0411090
$201$	0	0
$202$	−1996.03	−0.695248
$203$	874.416	0.302325
$204$	0	0
$205$	2009.93	0.684777
$206$	1475.83	0.499155
$207$	0	0
$208$	−1467.72	−0.489271
$209$	3529.59	1.16817
$210$	0	0
$211$	1956.17	0.638237	0.319119	−	0.947715i	$-0.396613\pi$
$211$	1956.17	0.638237	0.319119	+	0.947715i	$0.396613\pi$
$212$	3620.91	1.17304
$213$	0	0
$214$	706.291	0.225612
$215$	−645.815	−0.204857
$216$	0	0
$217$	2063.54	0.645541
$218$	720.747	0.223923
$219$	0	0
$220$	−4635.63	−1.42061
$221$	−4990.10	−1.51887
$222$	0	0
$223$	−2468.88	−0.741383	−0.370691	−	0.928756i	$-0.620879\pi$
$223$	−2468.88	−0.741383	−0.370691	+	0.928756i	$0.620879\pi$
$224$	−1162.14	−0.346646
$225$	0	0
$226$	−781.159	−0.229920
$227$	151.185	0.0442049	0.0221025	−	0.999756i	$-0.492964\pi$
$227$	151.185	0.0442049	0.0221025	+	0.999756i	$0.492964\pi$
$228$	0	0
$229$	1267.37	0.365722	0.182861	−	0.983139i	$-0.441464\pi$
$229$	1267.37	0.365722	0.182861	+	0.983139i	$0.441464\pi$
$230$	278.444	0.0798264
$231$	0	0
$232$	1957.22	0.553871
$233$	4649.22	1.30721	0.653606	−	0.756835i	$-0.273255\pi$
$233$	4649.22	1.30721	0.653606	+	0.756835i	$0.273255\pi$
$234$	0	0
$235$	−2951.18	−0.819207
$236$	242.682	0.0669376
$237$	0	0
$238$	−968.011	−0.263642
$239$	4276.00	1.15729	0.578643	−	0.815581i	$-0.303583\pi$
$239$	4276.00	1.15729	0.578643	+	0.815581i	$0.303583\pi$
$240$	0	0
$241$	5841.28	1.56129	0.780643	−	0.624977i	$-0.214892\pi$
$241$	5841.28	1.56129	0.780643	+	0.624977i	$0.214892\pi$
$242$	−2192.67	−0.582438
$243$	0	0
$244$	−3147.28	−0.825754
$245$	563.864	0.147037
$246$	0	0
$247$	−2292.85	−0.590649
$248$	4618.87	1.18266
$249$	0	0
$250$	1423.44	0.360106
$251$	−6331.37	−1.59216	−0.796080	−	0.605191i	$-0.793097\pi$
$251$	−6331.37	−1.59216	−0.796080	+	0.605191i	$0.793097\pi$
$252$	0	0
$253$	−1344.11	−0.334007
$254$	2322.62	0.573757
$255$	0	0
$256$	−469.195	−0.114549
$257$	−2019.37	−0.490136	−0.245068	−	0.969506i	$-0.578810\pi$
$257$	−2019.37	−0.490136	−0.245068	+	0.969506i	$0.578810\pi$
$258$	0	0
$259$	−305.511	−0.0732954
$260$	3011.34	0.718289
$261$	0	0
$262$	1003.85	0.236709
$263$	−4394.67	−1.03037	−0.515184	−	0.857079i	$-0.672276\pi$
$263$	−4394.67	−1.03037	−0.515184	+	0.857079i	$0.672276\pi$
$264$	0	0
$265$	−6044.70	−1.40122
$266$	−444.781	−0.102524
$267$	0	0
$268$	−4209.94	−0.959564
$269$	−1062.63	−0.240855	−0.120427	−	0.992722i	$-0.538427\pi$
$269$	−1062.63	−0.240855	−0.120427	+	0.992722i	$0.538427\pi$
$270$	0	0
$271$	3365.24	0.754330	0.377165	−	0.926146i	$-0.376899\pi$
$271$	3365.24	0.754330	0.377165	+	0.926146i	$0.376899\pi$
$272$	5082.01	1.13288
$273$	0	0
$274$	993.407	0.219029
$275$	433.681	0.0950980
$276$	0	0
$277$	−927.202	−0.201120	−0.100560	−	0.994931i	$-0.532063\pi$
$277$	−927.202	−0.201120	−0.100560	+	0.994931i	$0.532063\pi$
$278$	−2865.92	−0.618298
$279$	0	0
$280$	1262.11	0.269377
$281$	3079.62	0.653790	0.326895	−	0.945061i	$-0.393998\pi$
$281$	3079.62	0.653790	0.326895	+	0.945061i	$0.393998\pi$
$282$	0	0
$283$	−7618.99	−1.60036	−0.800180	−	0.599760i	$-0.795263\pi$
$283$	−7618.99	−1.60036	−0.800180	+	0.599760i	$0.795263\pi$
$284$	−2741.17	−0.572742
$285$	0	0
$286$	2333.99	0.482559
$287$	1222.64	0.251465
$288$	0	0
$289$	12365.3	2.51685
$290$	−1512.27	−0.306220
$291$	0	0
$292$	3105.52	0.622386
$293$	9123.10	1.81903	0.909517	−	0.415666i	$-0.136451\pi$
$293$	9123.10	1.81903	0.909517	+	0.415666i	$0.136451\pi$
$294$	0	0
$295$	−405.131	−0.0799580
$296$	−683.831	−0.134280
$297$	0	0
$298$	−2212.90	−0.430168
$299$	873.146	0.168881
$300$	0	0
$301$	−392.851	−0.0752278
$302$	−105.853	−0.0201693
$303$	0	0
$304$	2335.08	0.440546
$305$	5254.03	0.986376
$306$	0	0
$307$	6385.34	1.18707	0.593535	−	0.804808i	$-0.297732\pi$
$307$	6385.34	1.18707	0.593535	+	0.804808i	$0.297732\pi$
$308$	−2819.86	−0.521677
$309$	0	0
$310$	−3568.83	−0.653858
$311$	−3798.57	−0.692595	−0.346298	−	0.938125i	$-0.612561\pi$
$311$	−3798.57	−0.692595	−0.346298	+	0.938125i	$0.612561\pi$
$312$	0	0
$313$	6875.34	1.24159	0.620794	−	0.783973i	$-0.286810\pi$
$313$	6875.34	1.24159	0.620794	+	0.783973i	$0.286810\pi$
$314$	−1154.64	−0.207517
$315$	0	0
$316$	6147.57	1.09439
$317$	1063.45	0.188420	0.0942101	−	0.995552i	$-0.469967\pi$
$317$	1063.45	0.188420	0.0942101	+	0.995552i	$0.469967\pi$
$318$	0	0
$319$	7300.09	1.28127
$320$	−1549.33	−0.270656
$321$	0	0
$322$	169.378	0.0293139
$323$	7939.01	1.36761
$324$	0	0
$325$	−281.722	−0.0480835
$326$	−836.874	−0.142178
$327$	0	0
$328$	2736.67	0.460693
$329$	−1795.21	−0.300830
$330$	0	0
$331$	−1052.20	−0.174725	−0.0873625	−	0.996177i	$-0.527844\pi$
$331$	−1052.20	−0.174725	−0.0873625	+	0.996177i	$0.527844\pi$
$332$	9003.49	1.48835
$333$	0	0
$334$	−3885.71	−0.636576
$335$	7028.02	1.14621
$336$	0	0
$337$	−4269.30	−0.690099	−0.345050	−	0.938584i	$-0.612138\pi$
$337$	−4269.30	−0.690099	−0.345050	+	0.938584i	$0.612138\pi$
$338$	795.153	0.127960
$339$	0	0
$340$	−10426.8	−1.66315
$341$	17227.6	2.73585
$342$	0	0
$343$	343.000	0.0539949
$344$	−879.327	−0.137820
$345$	0	0
$346$	2237.72	0.347690
$347$	−10070.4	−1.55794	−0.778970	−	0.627061i	$-0.784257\pi$
$347$	−10070.4	−1.55794	−0.778970	+	0.627061i	$0.784257\pi$
$348$	0	0
$349$	−8099.84	−1.24233	−0.621167	−	0.783678i	$-0.713341\pi$
$349$	−8099.84	−1.24233	−0.621167	+	0.783678i	$0.713341\pi$
$350$	−54.6502	−0.00834622
$351$	0	0
$352$	−9702.17	−1.46911
$353$	−2527.31	−0.381063	−0.190532	−	0.981681i	$-0.561021\pi$
$353$	−2527.31	−0.381063	−0.190532	+	0.981681i	$0.561021\pi$
$354$	0	0
$355$	4576.08	0.684149
$356$	−861.928	−0.128320
$357$	0	0
$358$	−1813.77	−0.267768
$359$	−5046.63	−0.741925	−0.370962	−	0.928648i	$-0.620972\pi$
$359$	−5046.63	−0.741925	−0.370962	+	0.928648i	$0.620972\pi$
$360$	0	0
$361$	−3211.19	−0.468172
$362$	−2620.20	−0.380428
$363$	0	0
$364$	1831.80	0.263771
$365$	−5184.31	−0.743450
$366$	0	0
$367$	6700.72	0.953065	0.476532	−	0.879157i	$-0.341893\pi$
$367$	6700.72	0.953065	0.476532	+	0.879157i	$0.341893\pi$
$368$	−889.228	−0.125963
$369$	0	0
$370$	528.371	0.0742397
$371$	−3677.00	−0.514557
$372$	0	0
$373$	1853.52	0.257296	0.128648	−	0.991690i	$-0.458936\pi$
$373$	1853.52	0.257296	0.128648	+	0.991690i	$0.458936\pi$
$374$	−8081.48	−1.11733
$375$	0	0
$376$	−4018.25	−0.551132
$377$	−4742.19	−0.647839
$378$	0	0
$379$	999.292	0.135436	0.0677179	−	0.997705i	$-0.478428\pi$
$379$	999.292	0.135436	0.0677179	+	0.997705i	$0.478428\pi$
$380$	−4790.89	−0.646756
$381$	0	0
$382$	−3026.16	−0.405319
$383$	10859.5	1.44881	0.724404	−	0.689376i	$-0.242115\pi$
$383$	10859.5	1.44881	0.724404	+	0.689376i	$0.242115\pi$
$384$	0	0
$385$	4707.44	0.623152
$386$	1767.74	0.233097
$387$	0	0
$388$	1239.08	0.162125
$389$	−5895.92	−0.768470	−0.384235	−	0.923235i	$-0.625535\pi$
$389$	−5895.92	−0.768470	−0.384235	+	0.923235i	$0.625535\pi$
$390$	0	0
$391$	−3023.28	−0.391033
$392$	767.744	0.0989207
$393$	0	0
$394$	4539.64	0.580467
$395$	−10262.7	−1.30727
$396$	0	0
$397$	12820.0	1.62070	0.810348	−	0.585949i	$-0.199278\pi$
$397$	12820.0	1.62070	0.810348	+	0.585949i	$0.199278\pi$
$398$	−2845.79	−0.358409
$399$	0	0
$400$	286.911	0.0358639
$401$	1357.45	0.169047	0.0845236	−	0.996421i	$-0.473063\pi$
$401$	1357.45	0.169047	0.0845236	+	0.996421i	$0.473063\pi$
$402$	0	0
$403$	−11191.1	−1.38330
$404$	−13078.5	−1.61059
$405$	0	0
$406$	−919.920	−0.112450
$407$	−2550.57	−0.310631
$408$	0	0
$409$	−4543.38	−0.549281	−0.274640	−	0.961547i	$-0.588559\pi$
$409$	−4543.38	−0.549281	−0.274640	+	0.961547i	$0.588559\pi$
$410$	−2114.52	−0.254704
$411$	0	0
$412$	9669.98	1.15633
$413$	−246.442	−0.0293623
$414$	0	0
$415$	−15030.3	−1.77785
$416$	6302.60	0.742813
$417$	0	0
$418$	−3713.27	−0.434502
$419$	−7976.02	−0.929961	−0.464981	−	0.885321i	$-0.653939\pi$
$419$	−7976.02	−0.929961	−0.464981	+	0.885321i	$0.653939\pi$
$420$	0	0
$421$	−2993.41	−0.346531	−0.173266	−	0.984875i	$-0.555432\pi$
$421$	−2993.41	−0.346531	−0.173266	+	0.984875i	$0.555432\pi$
$422$	−2057.96	−0.237394
$423$	0	0
$424$	−8230.32	−0.942687
$425$	975.466	0.111334
$426$	0	0
$427$	3196.04	0.362218
$428$	4627.79	0.522646
$429$	0	0
$430$	679.423	0.0761970
$431$	6123.56	0.684365	0.342183	−	0.939634i	$-0.388834\pi$
$431$	6123.56	0.684365	0.342183	+	0.939634i	$0.388834\pi$
$432$	0	0
$433$	11225.8	1.24591	0.622953	−	0.782260i	$-0.285933\pi$
$433$	11225.8	1.24591	0.622953	+	0.782260i	$0.285933\pi$
$434$	−2170.93	−0.240110
$435$	0	0
$436$	4722.51	0.518732
$437$	−1389.13	−0.152062
$438$	0	0
$439$	15393.0	1.67350	0.836750	−	0.547585i	$-0.184453\pi$
$439$	15393.0	1.67350	0.836750	+	0.547585i	$0.184453\pi$
$440$	10536.8	1.14164
$441$	0	0
$442$	5249.78	0.564947
$443$	−8429.13	−0.904018	−0.452009	−	0.892013i	$-0.649293\pi$
$443$	−8429.13	−0.904018	−0.452009	+	0.892013i	$0.649293\pi$
$444$	0	0
$445$	1438.89	0.153281
$446$	2597.36	0.275759
$447$	0	0
$448$	−942.460	−0.0993907
$449$	−16006.3	−1.68237	−0.841186	−	0.540746i	$-0.818142\pi$
$449$	−16006.3	−1.68237	−0.841186	+	0.540746i	$0.818142\pi$
$450$	0	0
$451$	10207.3	1.06573
$452$	−5118.34	−0.532625
$453$	0	0
$454$	−159.053	−0.0164421
$455$	−3057.99	−0.315078
$456$	0	0
$457$	−12620.9	−1.29186	−0.645930	−	0.763397i	$-0.723530\pi$
$457$	−12620.9	−1.29186	−0.645930	+	0.763397i	$0.723530\pi$
$458$	−1333.32	−0.136031
$459$	0	0
$460$	1824.43	0.184923
$461$	3587.54	0.362448	0.181224	−	0.983442i	$-0.441994\pi$
$461$	3587.54	0.362448	0.181224	+	0.983442i	$0.441994\pi$
$462$	0	0
$463$	6037.85	0.606054	0.303027	−	0.952982i	$-0.402003\pi$
$463$	6037.85	0.606054	0.303027	+	0.952982i	$0.402003\pi$
$464$	4829.53	0.483201
$465$	0	0
$466$	−4891.16	−0.486220
$467$	−15082.1	−1.49447	−0.747233	−	0.664562i	$-0.768618\pi$
$467$	−15082.1	−1.49447	−0.747233	+	0.664562i	$0.768618\pi$
$468$	0	0
$469$	4275.16	0.420914
$470$	3104.76	0.304706
$471$	0	0
$472$	−551.616	−0.0537928
$473$	−3279.73	−0.318821
$474$	0	0
$475$	448.206	0.0432950
$476$	−6342.64	−0.610745
$477$	0	0
$478$	−4498.52	−0.430455
$479$	1315.02	0.125438	0.0627190	−	0.998031i	$-0.480023\pi$
$479$	1315.02	0.125438	0.0627190	+	0.998031i	$0.480023\pi$
$480$	0	0
$481$	1656.87	0.157062
$482$	−6145.26	−0.580724
$483$	0	0
$484$	−14366.9	−1.34926
$485$	−2068.50	−0.193661
$486$	0	0
$487$	5647.77	0.525513	0.262757	−	0.964862i	$-0.415368\pi$
$487$	5647.77	0.525513	0.262757	+	0.964862i	$0.415368\pi$
$488$	7153.76	0.663597
$489$	0	0
$490$	−593.207	−0.0546906
$491$	19855.6	1.82499	0.912493	−	0.409091i	$-0.134154\pi$
$491$	19855.6	1.82499	0.912493	+	0.409091i	$0.134154\pi$
$492$	0	0
$493$	16419.9	1.50003
$494$	2412.16	0.219693
$495$	0	0
$496$	11397.3	1.03176
$497$	2783.64	0.251234
$498$	0	0
$499$	−11353.9	−1.01858	−0.509291	−	0.860595i	$-0.670092\pi$
$499$	−11353.9	−1.01858	−0.509291	+	0.860595i	$0.670092\pi$
$500$	9326.74	0.834209
$501$	0	0
$502$	6660.85	0.592208
$503$	−15121.3	−1.34041	−0.670205	−	0.742176i	$-0.733794\pi$
$503$	−15121.3	−1.34041	−0.670205	+	0.742176i	$0.733794\pi$
$504$	0	0
$505$	21833.0	1.92387
$506$	1414.06	0.124235
$507$	0	0
$508$	15218.4	1.32915
$509$	−4819.60	−0.419696	−0.209848	−	0.977734i	$-0.567297\pi$
$509$	−4819.60	−0.419696	−0.209848	+	0.977734i	$0.567297\pi$
$510$	0	0
$511$	−3153.63	−0.273010
$512$	−11264.8	−0.972342
$513$	0	0
$514$	2124.46	0.182307
$515$	−16142.9	−1.38125
$516$	0	0
$517$	−14987.4	−1.27494
$518$	321.409	0.0272624
$519$	0	0
$520$	−6844.75	−0.577235
$521$	−9010.06	−0.757655	−0.378827	−	0.925467i	$-0.623673\pi$
$521$	−9010.06	−0.757655	−0.378827	+	0.925467i	$0.623673\pi$
$522$	0	0
$523$	4230.86	0.353734	0.176867	−	0.984235i	$-0.443404\pi$
$523$	4230.86	0.353734	0.176867	+	0.984235i	$0.443404\pi$
$524$	6577.44	0.548353
$525$	0	0
$526$	4623.36	0.383248
$527$	38749.5	3.20295
$528$	0	0
$529$	529.000	0.0434783
$530$	6359.26	0.521186
$531$	0	0
$532$	−2914.31	−0.237502
$533$	−6630.72	−0.538852
$534$	0	0
$535$	−7725.56	−0.624308
$536$	9569.19	0.771130
$537$	0	0
$538$	1117.93	0.0895865
$539$	2863.55	0.228834
$540$	0	0
$541$	1109.88	0.0882021	0.0441011	−	0.999027i	$-0.485958\pi$
$541$	1109.88	0.0882021	0.0441011	+	0.999027i	$0.485958\pi$
$542$	−3540.36	−0.280575
$543$	0	0
$544$	−21822.8	−1.71994
$545$	−7883.69	−0.619633
$546$	0	0
$547$	632.641	0.0494511	0.0247256	−	0.999694i	$-0.492129\pi$
$547$	632.641	0.0494511	0.0247256	+	0.999694i	$0.492129\pi$
$548$	6509.04	0.507395
$549$	0	0
$550$	−456.249	−0.0353719
$551$	7544.59	0.583322
$552$	0	0
$553$	−6242.80	−0.480056
$554$	975.453	0.0748069
$555$	0	0
$556$	−18778.2	−1.43233
$557$	−3748.00	−0.285112	−0.142556	−	0.989787i	$-0.545532\pi$
$557$	−3748.00	−0.285112	−0.142556	+	0.989787i	$0.545532\pi$
$558$	0	0
$559$	2130.54	0.161202
$560$	3114.31	0.235006
$561$	0	0
$562$	−3239.89	−0.243179
$563$	18848.1	1.41093	0.705463	−	0.708747i	$-0.250739\pi$
$563$	18848.1	1.41093	0.705463	+	0.708747i	$0.250739\pi$
$564$	0	0
$565$	8544.49	0.636229
$566$	8015.48	0.595257
$567$	0	0
$568$	6230.68	0.460270
$569$	−6473.34	−0.476936	−0.238468	−	0.971150i	$-0.576645\pi$
$569$	−6473.34	−0.476936	−0.238468	+	0.971150i	$0.576645\pi$
$570$	0	0
$571$	−2488.64	−0.182393	−0.0911965	−	0.995833i	$-0.529069\pi$
$571$	−2488.64	−0.182393	−0.0911965	+	0.995833i	$0.529069\pi$
$572$	15292.9	1.11788
$573$	0	0
$574$	−1286.27	−0.0935328
$575$	−170.683	−0.0123791
$576$	0	0
$577$	1719.46	0.124059	0.0620295	−	0.998074i	$-0.480243\pi$
$577$	1719.46	0.124059	0.0620295	+	0.998074i	$0.480243\pi$
$578$	−13008.8	−0.936148
$579$	0	0
$580$	−9908.78	−0.709378
$581$	−9142.97	−0.652864
$582$	0	0
$583$	−30697.6	−2.18073
$584$	−7058.84	−0.500166
$585$	0	0
$586$	−9597.86	−0.676594
$587$	4663.30	0.327896	0.163948	−	0.986469i	$-0.447577\pi$
$587$	4663.30	0.327896	0.163948	+	0.986469i	$0.447577\pi$
$588$	0	0
$589$	17804.6	1.24554
$590$	426.213	0.0297405
$591$	0	0
$592$	−1687.38	−0.117147
$593$	−13086.8	−0.906259	−0.453129	−	0.891445i	$-0.649692\pi$
$593$	−13086.8	−0.906259	−0.453129	+	0.891445i	$0.649692\pi$
$594$	0	0
$595$	10588.3	0.729544
$596$	−14499.5	−0.996512
$597$	0	0
$598$	−918.584	−0.0628155
$599$	13990.5	0.954320	0.477160	−	0.878816i	$-0.341666\pi$
$599$	13990.5	0.954320	0.477160	+	0.878816i	$0.341666\pi$
$600$	0	0
$601$	−26603.8	−1.80565	−0.902823	−	0.430013i	$-0.858509\pi$
$601$	−26603.8	−1.80565	−0.902823	+	0.430013i	$0.858509\pi$
$602$	413.295	0.0279811
$603$	0	0
$604$	−693.571	−0.0467235
$605$	23983.9	1.61171
$606$	0	0
$607$	24907.8	1.66553	0.832766	−	0.553625i	$-0.186756\pi$
$607$	24907.8	1.66553	0.832766	+	0.553625i	$0.186756\pi$
$608$	−10027.1	−0.668838
$609$	0	0
$610$	−5527.44	−0.366885
$611$	9735.90	0.644635
$612$	0	0
$613$	17527.2	1.15484	0.577419	−	0.816448i	$-0.304060\pi$
$613$	17527.2	1.15484	0.577419	+	0.816448i	$0.304060\pi$
$614$	−6717.62	−0.441533
$615$	0	0
$616$	6409.54	0.419233
$617$	−6259.53	−0.408426	−0.204213	−	0.978926i	$-0.565464\pi$
$617$	−6259.53	−0.408426	−0.204213	+	0.978926i	$0.565464\pi$
$618$	0	0
$619$	−6671.61	−0.433206	−0.216603	−	0.976260i	$-0.569498\pi$
$619$	−6671.61	−0.433206	−0.216603	+	0.976260i	$0.569498\pi$
$620$	−23383.8	−1.51470
$621$	0	0
$622$	3996.24	0.257612
$623$	875.280	0.0562879
$624$	0	0
$625$	−16497.6	−1.05584
$626$	−7233.13	−0.461812
$627$	0	0
$628$	−7565.49	−0.480726
$629$	−5736.92	−0.363666
$630$	0	0
$631$	−19637.6	−1.23892	−0.619460	−	0.785028i	$-0.712648\pi$
$631$	−19637.6	−1.23892	−0.619460	+	0.785028i	$0.712648\pi$
$632$	−13973.4	−0.879481
$633$	0	0
$634$	−1118.79	−0.0700833
$635$	−25405.4	−1.58769
$636$	0	0
$637$	−1860.18	−0.115703
$638$	−7679.98	−0.476573
$639$	0	0
$640$	16913.7	1.04464
$641$	−19448.6	−1.19840	−0.599200	−	0.800600i	$-0.704514\pi$
$641$	−19448.6	−1.19840	−0.599200	+	0.800600i	$0.704514\pi$
$642$	0	0
$643$	−29409.2	−1.80371	−0.901857	−	0.432036i	$-0.857795\pi$
$643$	−29409.2	−1.80371	−0.901857	+	0.432036i	$0.857795\pi$
$644$	1109.81	0.0679077
$645$	0	0
$646$	−8352.15	−0.508685
$647$	−3249.78	−0.197468	−0.0987342	−	0.995114i	$-0.531479\pi$
$647$	−3249.78	−0.197468	−0.0987342	+	0.995114i	$0.531479\pi$
$648$	0	0
$649$	−2057.43	−0.124439
$650$	296.383	0.0178847
$651$	0	0
$652$	−5483.40	−0.329366
$653$	−24372.7	−1.46061	−0.730304	−	0.683122i	$-0.760622\pi$
$653$	−24372.7	−1.46061	−0.730304	+	0.683122i	$0.760622\pi$
$654$	0	0
$655$	−10980.3	−0.655016
$656$	6752.85	0.401912
$657$	0	0
$658$	1888.63	0.111894
$659$	10976.6	0.648844	0.324422	−	0.945912i	$-0.394830\pi$
$659$	10976.6	0.648844	0.324422	+	0.945912i	$0.394830\pi$
$660$	0	0
$661$	−9058.02	−0.533005	−0.266502	−	0.963834i	$-0.585868\pi$
$661$	−9058.02	−0.533005	−0.266502	+	0.963834i	$0.585868\pi$
$662$	1106.95	0.0649894
$663$	0	0
$664$	−20464.9	−1.19607
$665$	4865.11	0.283700
$666$	0	0
$667$	−2873.08	−0.166786
$668$	−25460.1	−1.47467
$669$	0	0
$670$	−7393.75	−0.426337
$671$	26682.2	1.53511
$672$	0	0
$673$	−29524.4	−1.69106	−0.845529	−	0.533930i	$-0.820715\pi$
$673$	−29524.4	−1.69106	−0.845529	+	0.533930i	$0.820715\pi$
$674$	4491.47	0.256684
$675$	0	0
$676$	5210.03	0.296429
$677$	5132.51	0.291372	0.145686	−	0.989331i	$-0.453461\pi$
$677$	5132.51	0.291372	0.145686	+	0.989331i	$0.453461\pi$
$678$	0	0
$679$	−1258.27	−0.0711165
$680$	23700.0	1.33655
$681$	0	0
$682$	−18124.1	−1.01761
$683$	13707.3	0.767927	0.383964	−	0.923348i	$-0.374559\pi$
$683$	13707.3	0.767927	0.383964	+	0.923348i	$0.374559\pi$
$684$	0	0
$685$	−10866.1	−0.606091
$686$	−360.850	−0.0200835
$687$	0	0
$688$	−2169.78	−0.120235
$689$	19941.4	1.10262
$690$	0	0
$691$	−12803.3	−0.704861	−0.352431	−	0.935838i	$-0.614645\pi$
$691$	−12803.3	−0.704861	−0.352431	+	0.935838i	$0.614645\pi$
$692$	14662.1	0.805447
$693$	0	0
$694$	10594.4	0.579479
$695$	31348.1	1.71094
$696$	0	0
$697$	22958.9	1.24768
$698$	8521.35	0.462089
$699$	0	0
$700$	−358.081	−0.0193346
$701$	−2657.29	−0.143173	−0.0715867	−	0.997434i	$-0.522806\pi$
$701$	−2657.29	−0.143173	−0.0715867	+	0.997434i	$0.522806\pi$
$702$	0	0
$703$	−2635.99	−0.141420
$704$	−7868.16	−0.421225
$705$	0	0
$706$	2658.84	0.141737
$707$	13281.1	0.706486
$708$	0	0
$709$	3616.06	0.191543	0.0957714	−	0.995403i	$-0.469468\pi$
$709$	3616.06	0.191543	0.0957714	+	0.995403i	$0.469468\pi$
$710$	−4814.21	−0.254471
$711$	0	0
$712$	1959.16	0.103122
$713$	−6780.21	−0.356130
$714$	0	0
$715$	−25529.7	−1.33532
$716$	−11884.3	−0.620302
$717$	0	0
$718$	5309.26	0.275961
$719$	7969.18	0.413352	0.206676	−	0.978409i	$-0.433735\pi$
$719$	7969.18	0.413352	0.206676	+	0.978409i	$0.433735\pi$
$720$	0	0
$721$	−9819.79	−0.507223
$722$	3378.30	0.174138
$723$	0	0
$724$	−17168.2	−0.881286
$725$	927.004	0.0474870
$726$	0	0
$727$	−22654.0	−1.15569	−0.577847	−	0.816145i	$-0.696107\pi$
$727$	−22654.0	−1.15569	−0.577847	+	0.816145i	$0.696107\pi$
$728$	−4163.68	−0.211973
$729$	0	0
$730$	5454.10	0.276528
$731$	−7377.01	−0.373254
$732$	0	0
$733$	16963.5	0.854792	0.427396	−	0.904064i	$-0.359431\pi$
$733$	16963.5	0.854792	0.427396	+	0.904064i	$0.359431\pi$
$734$	−7049.42	−0.354494
$735$	0	0
$736$	3818.46	0.191237
$737$	35691.3	1.78386
$738$	0	0
$739$	−28555.4	−1.42142	−0.710709	−	0.703486i	$-0.751626\pi$
$739$	−28555.4	−1.42142	−0.710709	+	0.703486i	$0.751626\pi$
$740$	3462.01	0.171981
$741$	0	0
$742$	3868.35	0.191390
$743$	36544.2	1.80441	0.902204	−	0.431309i	$-0.141948\pi$
$743$	36544.2	1.80441	0.902204	+	0.431309i	$0.141948\pi$
$744$	0	0
$745$	24205.2	1.19035
$746$	−1949.97	−0.0957019
$747$	0	0
$748$	−52951.8	−2.58838
$749$	−4699.48	−0.229259
$750$	0	0
$751$	27019.0	1.31283	0.656417	−	0.754398i	$-0.272071\pi$
$751$	27019.0	1.31283	0.656417	+	0.754398i	$0.272071\pi$
$752$	−9915.22	−0.480812
$753$	0	0
$754$	4988.97	0.240965
$755$	1157.84	0.0558120
$756$	0	0
$757$	28132.1	1.35070	0.675348	−	0.737499i	$-0.263993\pi$
$757$	28132.1	1.35070	0.675348	+	0.737499i	$0.263993\pi$
$758$	−1051.29	−0.0503756
$759$	0	0
$760$	10889.7	0.519750
$761$	9242.97	0.440286	0.220143	−	0.975468i	$-0.429348\pi$
$761$	9242.97	0.440286	0.220143	+	0.975468i	$0.429348\pi$
$762$	0	0
$763$	−4795.67	−0.227542
$764$	−19828.1	−0.938947
$765$	0	0
$766$	−11424.6	−0.538887
$767$	1336.52	0.0629191
$768$	0	0
$769$	6057.93	0.284076	0.142038	−	0.989861i	$-0.454634\pi$
$769$	6057.93	0.284076	0.142038	+	0.989861i	$0.454634\pi$
$770$	−4952.41	−0.231783
$771$	0	0
$772$	11582.6	0.539984
$773$	−25040.9	−1.16515	−0.582575	−	0.812777i	$-0.697955\pi$
$773$	−25040.9	−1.16515	−0.582575	+	0.812777i	$0.697955\pi$
$774$	0	0
$775$	2187.65	0.101397
$776$	−2816.42	−0.130288
$777$	0	0
$778$	6202.74	0.285834
$779$	10549.1	0.485189
$780$	0	0
$781$	23239.3	1.06475
$782$	3180.61	0.145445
$783$	0	0
$784$	1894.44	0.0862993
$785$	12629.7	0.574235
$786$	0	0
$787$	5226.71	0.236737	0.118369	−	0.992970i	$-0.462234\pi$
$787$	5226.71	0.236737	0.118369	+	0.992970i	$0.462234\pi$
$788$	29744.8	1.34469
$789$	0	0
$790$	10796.7	0.486241
$791$	5197.63	0.233637
$792$	0	0
$793$	−17333.0	−0.776181
$794$	−13487.1	−0.602821
$795$	0	0
$796$	−18646.3	−0.830278
$797$	−42939.3	−1.90839	−0.954195	−	0.299184i	$-0.903286\pi$
$797$	−42939.3	−1.90839	−0.954195	+	0.299184i	$0.903286\pi$
$798$	0	0
$799$	−33710.7	−1.49261
$800$	−1232.03	−0.0544487
$801$	0	0
$802$	−1428.09	−0.0628774
$803$	−26328.2	−1.15704
$804$	0	0
$805$	−1852.70	−0.0811167
$806$	11773.5	0.514522
$807$	0	0
$808$	29727.3	1.29431
$809$	−24276.4	−1.05502	−0.527512	−	0.849548i	$-0.676875\pi$
$809$	−24276.4	−1.05502	−0.527512	+	0.849548i	$0.676875\pi$
$810$	0	0
$811$	−40151.4	−1.73848	−0.869238	−	0.494393i	$-0.835390\pi$
$811$	−40151.4	−1.73848	−0.869238	+	0.494393i	$0.835390\pi$
$812$	−6027.53	−0.260499
$813$	0	0
$814$	2683.30	0.115540
$815$	9153.90	0.393432
$816$	0	0
$817$	−3389.58	−0.145149
$818$	4779.82	0.204306
$819$	0	0
$820$	−13854.8	−0.590039
$821$	36686.1	1.55951	0.779753	−	0.626087i	$-0.215345\pi$
$821$	36686.1	1.55951	0.779753	+	0.626087i	$0.215345\pi$
$822$	0	0
$823$	−26221.3	−1.11059	−0.555297	−	0.831652i	$-0.687395\pi$
$823$	−26221.3	−1.11059	−0.555297	+	0.831652i	$0.687395\pi$
$824$	−21979.8	−0.929252
$825$	0	0
$826$	259.267	0.0109214
$827$	−14824.5	−0.623337	−0.311669	−	0.950191i	$-0.600888\pi$
$827$	−14824.5	−0.623337	−0.311669	+	0.950191i	$0.600888\pi$
$828$	0	0
$829$	11740.0	0.491853	0.245927	−	0.969288i	$-0.420908\pi$
$829$	11740.0	0.491853	0.245927	+	0.969288i	$0.420908\pi$
$830$	15812.5	0.661276
$831$	0	0
$832$	5111.21	0.212980
$833$	6440.90	0.267904
$834$	0	0
$835$	42502.7	1.76152
$836$	−24330.2	−1.00655
$837$	0	0
$838$	8391.08	0.345901
$839$	−20270.8	−0.834118	−0.417059	−	0.908879i	$-0.636939\pi$
$839$	−20270.8	−0.834118	−0.417059	+	0.908879i	$0.636939\pi$
$840$	0	0
$841$	−8784.86	−0.360198
$842$	3149.18	0.128893
$843$	0	0
$844$	−13484.3	−0.549938
$845$	−8697.56	−0.354089
$846$	0	0
$847$	14589.4	0.591853
$848$	−20308.7	−0.822409
$849$	0	0
$850$	−1026.23	−0.0414110
$851$	1003.82	0.0404354
$852$	0	0
$853$	−13905.6	−0.558169	−0.279085	−	0.960267i	$-0.590031\pi$
$853$	−13905.6	−0.558169	−0.279085	+	0.960267i	$0.590031\pi$
$854$	−3362.36	−0.134728
$855$	0	0
$856$	−10518.9	−0.420012
$857$	10316.6	0.411212	0.205606	−	0.978635i	$-0.434083\pi$
$857$	10316.6	0.411212	0.205606	+	0.978635i	$0.434083\pi$
$858$	0	0
$859$	18090.8	0.718569	0.359285	−	0.933228i	$-0.383021\pi$
$859$	18090.8	0.718569	0.359285	+	0.933228i	$0.383021\pi$
$860$	4451.74	0.176515
$861$	0	0
$862$	−6442.22	−0.254551
$863$	−11751.7	−0.463536	−0.231768	−	0.972771i	$-0.574451\pi$
$863$	−11751.7	−0.463536	−0.231768	+	0.972771i	$0.574451\pi$
$864$	0	0
$865$	−24476.7	−0.962119
$866$	−11810.0	−0.463417
$867$	0	0
$868$	−14224.4	−0.556232
$869$	−52118.3	−2.03451
$870$	0	0
$871$	−23185.3	−0.901958
$872$	−10734.2	−0.416866
$873$	0	0
$874$	1461.42	0.0565599
$875$	−9471.23	−0.365927
$876$	0	0
$877$	25359.8	0.976443	0.488221	−	0.872720i	$-0.337646\pi$
$877$	25359.8	0.976443	0.488221	+	0.872720i	$0.337646\pi$
$878$	−16194.0	−0.622462
$879$	0	0
$880$	25999.9	0.995974
$881$	−2656.81	−0.101601	−0.0508004	−	0.998709i	$-0.516177\pi$
$881$	−2656.81	−0.101601	−0.0508004	+	0.998709i	$0.516177\pi$
$882$	0	0
$883$	3369.55	0.128419	0.0642097	−	0.997936i	$-0.479547\pi$
$883$	3369.55	0.128419	0.0642097	+	0.997936i	$0.479547\pi$
$884$	34397.8	1.30874
$885$	0	0
$886$	8867.78	0.336252
$887$	−1284.23	−0.0486135	−0.0243068	−	0.999705i	$-0.507738\pi$
$887$	−1284.23	−0.0486135	−0.0243068	+	0.999705i	$0.507738\pi$
$888$	0	0
$889$	−15454.1	−0.583032
$890$	−1513.77	−0.0570131
$891$	0	0
$892$	17018.5	0.638814
$893$	−15489.3	−0.580438
$894$	0	0
$895$	19839.5	0.740961
$896$	10288.6	0.383615
$897$	0	0
$898$	16839.3	0.625762
$899$	36824.4	1.36614
$900$	0	0
$901$	−69047.2	−2.55305
$902$	−10738.5	−0.396399
$903$	0	0
$904$	11634.0	0.428031
$905$	28660.3	1.05271
$906$	0	0
$907$	29573.4	1.08266	0.541329	−	0.840811i	$-0.317921\pi$
$907$	29573.4	1.08266	0.541329	+	0.840811i	$0.317921\pi$
$908$	−1042.15	−0.0380892
$909$	0	0
$910$	3217.12	0.117194
$911$	−49169.5	−1.78821	−0.894105	−	0.447857i	$-0.852187\pi$
$911$	−49169.5	−1.78821	−0.894105	+	0.447857i	$0.852187\pi$
$912$	0	0
$913$	−76330.4	−2.76689
$914$	13277.7	0.480510
$915$	0	0
$916$	−8736.26	−0.315125
$917$	−6679.34	−0.240536
$918$	0	0
$919$	47639.6	1.70999	0.854997	−	0.518633i	$-0.173559\pi$
$919$	47639.6	1.70999	0.854997	+	0.518633i	$0.173559\pi$
$920$	−4146.93	−0.148609
$921$	0	0
$922$	−3774.24	−0.134813
$923$	−15096.4	−0.538358
$924$	0	0
$925$	−323.885	−0.0115127
$926$	−6352.06	−0.225423
$927$	0	0
$928$	−20738.6	−0.733598
$929$	−33725.1	−1.19105	−0.595524	−	0.803338i	$-0.703055\pi$
$929$	−33725.1	−1.19105	−0.595524	+	0.803338i	$0.703055\pi$
$930$	0	0
$931$	2959.46	0.104181
$932$	−32048.0	−1.12636
$933$	0	0
$934$	15867.0	0.555870
$935$	88396.9	3.09186
$936$	0	0
$937$	−1257.44	−0.0438407	−0.0219204	−	0.999760i	$-0.506978\pi$
$937$	−1257.44	−0.0438407	−0.0219204	+	0.999760i	$0.506978\pi$
$938$	−4497.64	−0.156560
$939$	0	0
$940$	20343.1	0.705871
$941$	−25458.3	−0.881952	−0.440976	−	0.897519i	$-0.645368\pi$
$941$	−25458.3	−0.881952	−0.440976	+	0.897519i	$0.645368\pi$
$942$	0	0
$943$	−4017.26	−0.138727
$944$	−1361.14	−0.0469293
$945$	0	0
$946$	3450.41	0.118586
$947$	10810.6	0.370959	0.185479	−	0.982648i	$-0.440616\pi$
$947$	10810.6	0.370959	0.185479	+	0.982648i	$0.440616\pi$
$948$	0	0
$949$	17103.0	0.585022
$950$	−471.530	−0.0161036
$951$	0	0
$952$	14416.8	0.490810
$953$	−10790.3	−0.366771	−0.183386	−	0.983041i	$-0.558706\pi$
$953$	−10790.3	−0.366771	−0.183386	+	0.983041i	$0.558706\pi$
$954$	0	0
$955$	33100.8	1.12159
$956$	−29475.3	−0.997177
$957$	0	0
$958$	−1383.45	−0.0466569
$959$	−6609.87	−0.222569
$960$	0	0
$961$	57111.2	1.91706
$962$	−1743.09	−0.0584194
$963$	0	0
$964$	−40265.2	−1.34528
$965$	−19335.9	−0.645019
$966$	0	0
$967$	31507.2	1.04778	0.523889	−	0.851786i	$-0.324481\pi$
$967$	31507.2	1.04778	0.523889	+	0.851786i	$0.324481\pi$
$968$	32655.8	1.08430
$969$	0	0
$970$	2176.14	0.0720327
$971$	4081.26	0.134885	0.0674427	−	0.997723i	$-0.478516\pi$
$971$	4081.26	0.134885	0.0674427	+	0.997723i	$0.478516\pi$
$972$	0	0
$973$	19069.1	0.628292
$974$	−5941.68	−0.195466
$975$	0	0
$976$	17652.2	0.578928
$977$	−21784.9	−0.713367	−0.356683	−	0.934225i	$-0.616092\pi$
$977$	−21784.9	−0.713367	−0.356683	+	0.934225i	$0.616092\pi$
$978$	0	0
$979$	7307.31	0.238552
$980$	−3886.84	−0.126694
$981$	0	0
$982$	−20888.8	−0.678808
$983$	23937.3	0.776683	0.388342	−	0.921515i	$-0.373048\pi$
$983$	23937.3	0.776683	0.388342	+	0.921515i	$0.373048\pi$
$984$	0	0
$985$	−49655.6	−1.60625
$986$	−17274.4	−0.557939
$987$	0	0
$988$	15805.1	0.508934
$989$	1290.80	0.0415014
$990$	0	0
$991$	1362.92	0.0436877	0.0218439	−	0.999761i	$-0.493046\pi$
$991$	1362.92	0.0436877	0.0218439	+	0.999761i	$0.493046\pi$
$992$	−48941.3	−1.56642
$993$	0	0
$994$	−2928.50	−0.0934470
$995$	31127.9	0.991779
$996$	0	0
$997$	401.958	0.0127685	0.00638423	−	0.999980i	$-0.497968\pi$
$997$	401.958	0.0127685	0.00638423	+	0.999980i	$0.497968\pi$
$998$	11944.8	0.378863
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.4.a.m.1.4		9
3.2	odd	2		483.4.a.f.1.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.4.a.f.1.6	✓	9	3.2	odd	2
1449.4.a.m.1.4		9	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:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q-1.05204 q^{2} -6.89321 q^{4} +11.5074 q^{5} +7.00000 q^{7} +15.6682 q^{8} +O(q^{10})\)	\(q-1.05204 q^{2} -6.89321 q^{4} +11.5074 q^{5} +7.00000 q^{7} +15.6682 q^{8} -12.1063 q^{10} +58.4398 q^{11} -37.9629 q^{13} -7.36428 q^{14} +38.6621 q^{16} +131.447 q^{17} +60.3971 q^{19} -79.3232 q^{20} -61.4809 q^{22} -23.0000 q^{23} +7.42099 q^{25} +39.9384 q^{26} -48.2525 q^{28} +124.917 q^{29} +294.792 q^{31} -166.020 q^{32} -138.287 q^{34} +80.5520 q^{35} -43.6444 q^{37} -63.5401 q^{38} +180.301 q^{40} +174.663 q^{41} -56.1216 q^{43} -402.838 q^{44} +24.1969 q^{46} -256.458 q^{47} +49.0000 q^{49} -7.80717 q^{50} +261.686 q^{52} -525.286 q^{53} +672.492 q^{55} +109.678 q^{56} -131.417 q^{58} -35.2060 q^{59} +456.577 q^{61} -310.133 q^{62} -134.637 q^{64} -436.855 q^{65} +610.737 q^{67} -906.091 q^{68} -84.7439 q^{70} +397.663 q^{71} -450.519 q^{73} +45.9156 q^{74} -416.330 q^{76} +409.078 q^{77} -891.829 q^{79} +444.901 q^{80} -183.753 q^{82} -1306.14 q^{83} +1512.62 q^{85} +59.0421 q^{86} +915.649 q^{88} +125.040 q^{89} -265.740 q^{91} +158.544 q^{92} +269.804 q^{94} +695.015 q^{95} -179.753 q^{97} -51.5499 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 42 q^{4} - 29 q^{5} + 63 q^{7} + 39 q^{8}+O(q^{10}) \) 9 * q + 42 * q^4 - 29 * q^5 + 63 * q^7 + 39 * q^8	\( 9 q + 42 q^{4} - 29 q^{5} + 63 q^{7} + 39 q^{8} - 55 q^{10} - 12 q^{11} + 199 q^{13} + 170 q^{16} - 116 q^{17} + 260 q^{19} - 324 q^{20} - 265 q^{22} - 207 q^{23} + 438 q^{25} + 270 q^{26} + 294 q^{28} + 107 q^{29} + 440 q^{31} + 802 q^{32} + 295 q^{34} - 203 q^{35} + 563 q^{37} + 569 q^{38} - 640 q^{40} - 243 q^{41} + 435 q^{43} - 1025 q^{44} + 133 q^{47} + 441 q^{49} - 104 q^{50} + 2693 q^{52} - 958 q^{53} + 1846 q^{55} + 273 q^{56} + 2796 q^{58} - 538 q^{59} + 1374 q^{61} - 1263 q^{62} - 83 q^{64} - 745 q^{65} + 752 q^{67} - 5593 q^{68} - 385 q^{70} + 418 q^{71} + 2406 q^{73} - 352 q^{74} + 2765 q^{76} - 84 q^{77} - 486 q^{79} - 5709 q^{80} + 2726 q^{82} - 106 q^{83} + 4130 q^{85} + 2576 q^{86} + 1270 q^{88} - 234 q^{89} + 1393 q^{91} - 966 q^{92} + 4967 q^{94} + 3074 q^{95} + 2409 q^{97}+O(q^{100}) \) 9 * q + 42 * q^4 - 29 * q^5 + 63 * q^7 + 39 * q^8 - 55 * q^10 - 12 * q^11 + 199 * q^13 + 170 * q^16 - 116 * q^17 + 260 * q^19 - 324 * q^20 - 265 * q^22 - 207 * q^23 + 438 * q^25 + 270 * q^26 + 294 * q^28 + 107 * q^29 + 440 * q^31 + 802 * q^32 + 295 * q^34 - 203 * q^35 + 563 * q^37 + 569 * q^38 - 640 * q^40 - 243 * q^41 + 435 * q^43 - 1025 * q^44 + 133 * q^47 + 441 * q^49 - 104 * q^50 + 2693 * q^52 - 958 * q^53 + 1846 * q^55 + 273 * q^56 + 2796 * q^58 - 538 * q^59 + 1374 * q^61 - 1263 * q^62 - 83 * q^64 - 745 * q^65 + 752 * q^67 - 5593 * q^68 - 385 * q^70 + 418 * q^71 + 2406 * q^73 - 352 * q^74 + 2765 * q^76 - 84 * q^77 - 486 * q^79 - 5709 * q^80 + 2726 * q^82 - 106 * q^83 + 4130 * q^85 + 2576 * q^86 + 1270 * q^88 - 234 * q^89 + 1393 * q^91 - 966 * q^92 + 4967 * q^94 + 3074 * q^95 + 2409 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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