LMFDB - Embedded newform 153.4.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [153,4,Mod(1,153)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(153, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("153.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 153 = 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	153.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:	$9.02729223088$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1506848.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 17x^{2} + 18 $ x^4 - 17*x^2 + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.98315$ of defining polynomial
Character	$\chi$	$=$	153.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.47680 q^{2} -5.81906 q^{4} +11.3381 q^{5} -28.3925 q^{7} -20.4080 q^{8} +O(q^{10})$	\(q+1.47680 q^{2} -5.81906 q^{4} +11.3381 q^{5} -28.3925 q^{7} -20.4080 q^{8} +16.7441 q^{10} -56.2218 q^{11} +24.3820 q^{13} -41.9301 q^{14} +16.4140 q^{16} +17.0000 q^{17} -34.0201 q^{19} -65.9771 q^{20} -83.0284 q^{22} -108.384 q^{23} +3.55252 q^{25} +36.0074 q^{26} +165.218 q^{28} -266.590 q^{29} -207.151 q^{31} +187.504 q^{32} +25.1056 q^{34} -321.917 q^{35} +380.126 q^{37} -50.2410 q^{38} -231.388 q^{40} +451.737 q^{41} +395.919 q^{43} +327.158 q^{44} -160.061 q^{46} -179.212 q^{47} +463.136 q^{49} +5.24637 q^{50} -141.880 q^{52} -184.363 q^{53} -637.449 q^{55} +579.435 q^{56} -393.700 q^{58} -151.001 q^{59} +59.0113 q^{61} -305.920 q^{62} +145.594 q^{64} +276.446 q^{65} -56.7668 q^{67} -98.9240 q^{68} -475.408 q^{70} +265.014 q^{71} -704.699 q^{73} +561.371 q^{74} +197.965 q^{76} +1596.28 q^{77} +509.385 q^{79} +186.103 q^{80} +667.125 q^{82} +122.105 q^{83} +192.748 q^{85} +584.693 q^{86} +1147.37 q^{88} -710.816 q^{89} -692.267 q^{91} +630.690 q^{92} -264.660 q^{94} -385.724 q^{95} -834.277 q^{97} +683.960 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8	$ 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8} - 2 q^{10} - 50 q^{11} + 26 q^{13} - 80 q^{14} + 138 q^{16} + 68 q^{17} + 34 q^{19} - 312 q^{20} - 254 q^{22} - 382 q^{23} + 138 q^{25} + 22 q^{26} + 52 q^{28} - 540 q^{29} - 356 q^{31} - 730 q^{32} - 68 q^{34} - 304 q^{35} - 404 q^{37} - 298 q^{38} + 332 q^{40} + 114 q^{41} + 570 q^{43} + 1368 q^{44} - 290 q^{46} - 496 q^{47} - 224 q^{49} + 1862 q^{50} - 1012 q^{52} - 92 q^{53} - 482 q^{55} + 1428 q^{56} + 1324 q^{58} + 48 q^{59} - 1036 q^{61} + 2564 q^{62} + 2898 q^{64} + 342 q^{65} + 812 q^{67} + 442 q^{68} + 152 q^{70} - 1044 q^{71} - 1212 q^{73} + 1444 q^{74} + 2268 q^{76} + 564 q^{77} + 488 q^{79} + 1000 q^{80} - 938 q^{82} - 1708 q^{83} - 374 q^{85} + 2446 q^{86} - 3868 q^{88} + 8 q^{89} + 716 q^{91} - 1356 q^{92} - 1224 q^{94} - 1010 q^{95} - 76 q^{97} + 1472 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8 - 2 * q^10 - 50 * q^11 + 26 * q^13 - 80 * q^14 + 138 * q^16 + 68 * q^17 + 34 * q^19 - 312 * q^20 - 254 * q^22 - 382 * q^23 + 138 * q^25 + 22 * q^26 + 52 * q^28 - 540 * q^29 - 356 * q^31 - 730 * q^32 - 68 * q^34 - 304 * q^35 - 404 * q^37 - 298 * q^38 + 332 * q^40 + 114 * q^41 + 570 * q^43 + 1368 * q^44 - 290 * q^46 - 496 * q^47 - 224 * q^49 + 1862 * q^50 - 1012 * q^52 - 92 * q^53 - 482 * q^55 + 1428 * q^56 + 1324 * q^58 + 48 * q^59 - 1036 * q^61 + 2564 * q^62 + 2898 * q^64 + 342 * q^65 + 812 * q^67 + 442 * q^68 + 152 * q^70 - 1044 * q^71 - 1212 * q^73 + 1444 * q^74 + 2268 * q^76 + 564 * q^77 + 488 * q^79 + 1000 * q^80 - 938 * q^82 - 1708 * q^83 - 374 * q^85 + 2446 * q^86 - 3868 * q^88 + 8 * q^89 + 716 * q^91 - 1356 * q^92 - 1224 * q^94 - 1010 * q^95 - 76 * q^97 + 1472 * q^98

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 $

$2$	1.47680	0.522128	0.261064	−	0.965322i	$-0.415927\pi$
$2$	1.47680	0.522128	0.261064	+	0.965322i	$0.415927\pi$
$3$	0	0
$3$	0	0
$4$	−5.81906	−0.727383
$5$	11.3381	1.01411	0.507055	−	0.861914i	$-0.330734\pi$
$5$	11.3381	1.01411	0.507055	+	0.861914i	$0.330734\pi$
$6$	0	0
$7$	−28.3925	−1.53305	−0.766526	−	0.642213i	$-0.778017\pi$
$7$	−28.3925	−1.53305	−0.766526	+	0.642213i	$0.778017\pi$
$8$	−20.4080	−0.901914
$9$	0	0
$10$	16.7441	0.529495
$11$	−56.2218	−1.54105	−0.770523	−	0.637412i	$-0.780005\pi$
$11$	−56.2218	−1.54105	−0.770523	+	0.637412i	$0.780005\pi$
$12$	0	0
$13$	24.3820	0.520181	0.260091	−	0.965584i	$-0.416248\pi$
$13$	24.3820	0.520181	0.260091	+	0.965584i	$0.416248\pi$
$14$	−41.9301	−0.800449
$15$	0	0
$16$	16.4140	0.256468
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	−34.0201	−0.410777	−0.205388	−	0.978681i	$-0.565846\pi$
$19$	−34.0201	−0.410777	−0.205388	+	0.978681i	$0.565846\pi$
$20$	−65.9771	−0.737646
$21$	0	0
$22$	−83.0284	−0.804623
$23$	−108.384	−0.982588	−0.491294	−	0.870994i	$-0.663476\pi$
$23$	−108.384	−0.982588	−0.491294	+	0.870994i	$0.663476\pi$
$24$	0	0
$25$	3.55252	0.0284202
$26$	36.0074	0.271601
$27$	0	0
$28$	165.218	1.11512
$29$	−266.590	−1.70705	−0.853527	−	0.521049i	$-0.825541\pi$
$29$	−266.590	−1.70705	−0.853527	+	0.521049i	$0.825541\pi$
$30$	0	0
$31$	−207.151	−1.20017	−0.600086	−	0.799935i	$-0.704867\pi$
$31$	−207.151	−1.20017	−0.600086	+	0.799935i	$0.704867\pi$
$32$	187.504	1.03582
$33$	0	0
$34$	25.1056	0.126635
$35$	−321.917	−1.55469
$36$	0	0
$37$	380.126	1.68898	0.844492	−	0.535568i	$-0.179903\pi$
$37$	380.126	1.68898	0.844492	+	0.535568i	$0.179903\pi$
$38$	−50.2410	−0.214478
$39$	0	0
$40$	−231.388	−0.914641
$41$	451.737	1.72072	0.860359	−	0.509689i	$-0.170239\pi$
$41$	451.737	1.72072	0.860359	+	0.509689i	$0.170239\pi$
$42$	0	0
$43$	395.919	1.40412	0.702059	−	0.712119i	$-0.252264\pi$
$43$	395.919	1.40412	0.702059	+	0.712119i	$0.252264\pi$
$44$	327.158	1.12093
$45$	0	0
$46$	−160.061	−0.513036
$47$	−179.212	−0.556186	−0.278093	−	0.960554i	$-0.589702\pi$
$47$	−179.212	−0.556186	−0.278093	+	0.960554i	$0.589702\pi$
$48$	0	0
$49$	463.136	1.35025
$50$	5.24637	0.0148390
$51$	0	0
$52$	−141.880	−0.378371
$53$	−184.363	−0.477814	−0.238907	−	0.971042i	$-0.576789\pi$
$53$	−184.363	−0.477814	−0.238907	+	0.971042i	$0.576789\pi$
$54$	0	0
$55$	−637.449	−1.56279
$56$	579.435	1.38268
$57$	0	0
$58$	−393.700	−0.891300
$59$	−151.001	−0.333197	−0.166599	−	0.986025i	$-0.553278\pi$
$59$	−151.001	−0.333197	−0.166599	+	0.986025i	$0.553278\pi$
$60$	0	0
$61$	59.0113	0.123863	0.0619313	−	0.998080i	$-0.480274\pi$
$61$	59.0113	0.123863	0.0619313	+	0.998080i	$0.480274\pi$
$62$	−305.920	−0.626644
$63$	0	0
$64$	145.594	0.284364
$65$	276.446	0.527521
$66$	0	0
$67$	−56.7668	−0.103510	−0.0517550	−	0.998660i	$-0.516481\pi$
$67$	−56.7668	−0.103510	−0.0517550	+	0.998660i	$0.516481\pi$
$68$	−98.9240	−0.176416
$69$	0	0
$70$	−475.408	−0.811744
$71$	265.014	0.442977	0.221488	−	0.975163i	$-0.428909\pi$
$71$	265.014	0.442977	0.221488	+	0.975163i	$0.428909\pi$
$72$	0	0
$73$	−704.699	−1.12985	−0.564924	−	0.825143i	$-0.691094\pi$
$73$	−704.699	−1.12985	−0.564924	+	0.825143i	$0.691094\pi$
$74$	561.371	0.881865
$75$	0	0
$76$	197.965	0.298792
$77$	1596.28	2.36251
$78$	0	0
$79$	509.385	0.725447	0.362724	−	0.931897i	$-0.381847\pi$
$79$	509.385	0.725447	0.362724	+	0.931897i	$0.381847\pi$
$80$	186.103	0.260087
$81$	0	0
$82$	667.125	0.898435
$83$	122.105	0.161479	0.0807395	−	0.996735i	$-0.474272\pi$
$83$	122.105	0.161479	0.0807395	+	0.996735i	$0.474272\pi$
$84$	0	0
$85$	192.748	0.245958
$86$	584.693	0.733129
$87$	0	0
$88$	1147.37	1.38989
$89$	−710.816	−0.846588	−0.423294	−	0.905992i	$-0.639126\pi$
$89$	−710.816	−0.846588	−0.423294	+	0.905992i	$0.639126\pi$
$90$	0	0
$91$	−692.267	−0.797465
$92$	630.690	0.714717
$93$	0	0
$94$	−264.660	−0.290400
$95$	−385.724	−0.416573
$96$	0	0
$97$	−834.277	−0.873278	−0.436639	−	0.899637i	$-0.643831\pi$
$97$	−834.277	−0.873278	−0.436639	+	0.899637i	$0.643831\pi$
$98$	683.960	0.705004
$99$	0	0
$100$	−20.6724	−0.0206724
$101$	−1607.86	−1.58404	−0.792019	−	0.610496i	$-0.790970\pi$
$101$	−1607.86	−1.58404	−0.792019	+	0.610496i	$0.790970\pi$
$102$	0	0
$103$	1581.08	1.51251	0.756256	−	0.654276i	$-0.227026\pi$
$103$	1581.08	1.51251	0.756256	+	0.654276i	$0.227026\pi$
$104$	−497.588	−0.469159
$105$	0	0
$106$	−272.267	−0.249480
$107$	−2077.88	−1.87734	−0.938672	−	0.344810i	$-0.887943\pi$
$107$	−2077.88	−1.87734	−0.938672	+	0.344810i	$0.887943\pi$
$108$	0	0
$109$	88.7600	0.0779970	0.0389985	−	0.999239i	$-0.487583\pi$
$109$	88.7600	0.0779970	0.0389985	+	0.999239i	$0.487583\pi$
$110$	−941.384	−0.815977
$111$	0	0
$112$	−466.034	−0.393179
$113$	−519.748	−0.432688	−0.216344	−	0.976317i	$-0.569413\pi$
$113$	−519.748	−0.432688	−0.216344	+	0.976317i	$0.569413\pi$
$114$	0	0
$115$	−1228.86	−0.996453
$116$	1551.30	1.24168
$117$	0	0
$118$	−222.998	−0.173972
$119$	−482.673	−0.371820
$120$	0	0
$121$	1829.89	1.37482
$122$	87.1479	0.0646721
$123$	0	0
$124$	1205.42	0.872985
$125$	−1376.98	−0.985289
$126$	0	0
$127$	−479.358	−0.334931	−0.167465	−	0.985878i	$-0.553558\pi$
$127$	−479.358	−0.334931	−0.167465	+	0.985878i	$0.553558\pi$
$128$	−1285.02	−0.887349
$129$	0	0
$130$	408.255	0.275433
$131$	1535.16	1.02387	0.511936	−	0.859024i	$-0.328929\pi$
$131$	1535.16	1.02387	0.511936	+	0.859024i	$0.328929\pi$
$132$	0	0
$133$	965.918	0.629742
$134$	−83.8332	−0.0540454
$135$	0	0
$136$	−346.936	−0.218746
$137$	−655.102	−0.408534	−0.204267	−	0.978915i	$-0.565481\pi$
$137$	−655.102	−0.408534	−0.204267	+	0.978915i	$0.565481\pi$
$138$	0	0
$139$	−2539.34	−1.54952	−0.774762	−	0.632253i	$-0.782130\pi$
$139$	−2539.34	−1.54952	−0.774762	+	0.632253i	$0.782130\pi$
$140$	1873.26	1.13085
$141$	0	0
$142$	391.372	0.231290
$143$	−1370.80	−0.801623
$144$	0	0
$145$	−3022.63	−1.73114
$146$	−1040.70	−0.589925
$147$	0	0
$148$	−2211.98	−1.22854
$149$	−1314.06	−0.722498	−0.361249	−	0.932469i	$-0.617650\pi$
$149$	−1314.06	−0.722498	−0.361249	+	0.932469i	$0.617650\pi$
$150$	0	0
$151$	1921.33	1.03547	0.517734	−	0.855542i	$-0.326776\pi$
$151$	1921.33	1.03547	0.517734	+	0.855542i	$0.326776\pi$
$152$	694.283	0.370485
$153$	0	0
$154$	2357.39	1.23353
$155$	−2348.70	−1.21711
$156$	0	0
$157$	−22.3766	−0.0113748	−0.00568742	−	0.999984i	$-0.501810\pi$
$157$	−22.3766	−0.0113748	−0.00568742	+	0.999984i	$0.501810\pi$
$158$	752.260	0.378776
$159$	0	0
$160$	2125.94	1.05044
$161$	3077.28	1.50636
$162$	0	0
$163$	1965.65	0.944548	0.472274	−	0.881452i	$-0.343433\pi$
$163$	1965.65	0.944548	0.472274	+	0.881452i	$0.343433\pi$
$164$	−2628.68	−1.25162
$165$	0	0
$166$	180.325	0.0843127
$167$	−1382.17	−0.640454	−0.320227	−	0.947341i	$-0.603759\pi$
$167$	−1382.17	−0.640454	−0.320227	+	0.947341i	$0.603759\pi$
$168$	0	0
$169$	−1602.52	−0.729412
$170$	284.650	0.128421
$171$	0	0
$172$	−2303.88	−1.02133
$173$	−34.0941	−0.0149834	−0.00749170	−	0.999972i	$-0.502385\pi$
$173$	−34.0941	−0.0149834	−0.00749170	+	0.999972i	$0.502385\pi$
$174$	0	0
$175$	−100.865	−0.0435696
$176$	−922.822	−0.395229
$177$	0	0
$178$	−1049.73	−0.442027
$179$	1132.44	0.472862	0.236431	−	0.971648i	$-0.424022\pi$
$179$	1132.44	0.472862	0.236431	+	0.971648i	$0.424022\pi$
$180$	0	0
$181$	−1194.23	−0.490423	−0.245212	−	0.969470i	$-0.578857\pi$
$181$	−1194.23	−0.490423	−0.245212	+	0.969470i	$0.578857\pi$
$182$	−1022.34	−0.416379
$183$	0	0
$184$	2211.89	0.886210
$185$	4309.91	1.71282
$186$	0	0
$187$	−955.771	−0.373759
$188$	1042.84	0.404560
$189$	0	0
$190$	−569.637	−0.217504
$191$	−329.387	−0.124783	−0.0623916	−	0.998052i	$-0.519873\pi$
$191$	−329.387	−0.124783	−0.0623916	+	0.998052i	$0.519873\pi$
$192$	0	0
$193$	−2430.11	−0.906337	−0.453168	−	0.891425i	$-0.649706\pi$
$193$	−2430.11	−0.906337	−0.453168	+	0.891425i	$0.649706\pi$
$194$	−1232.06	−0.455963
$195$	0	0
$196$	−2695.02	−0.982149
$197$	−2352.42	−0.850776	−0.425388	−	0.905011i	$-0.639862\pi$
$197$	−2352.42	−0.850776	−0.425388	+	0.905011i	$0.639862\pi$
$198$	0	0
$199$	−1392.78	−0.496137	−0.248068	−	0.968743i	$-0.579796\pi$
$199$	−1392.78	−0.496137	−0.248068	+	0.968743i	$0.579796\pi$
$200$	−72.4999	−0.0256326
$201$	0	0
$202$	−2374.49	−0.827070
$203$	7569.17	2.61700
$204$	0	0
$205$	5121.84	1.74500
$206$	2334.94	0.789725
$207$	0	0
$208$	400.205	0.133410
$209$	1912.67	0.633026
$210$	0	0
$211$	3717.70	1.21297	0.606485	−	0.795095i	$-0.292579\pi$
$211$	3717.70	1.21297	0.606485	+	0.795095i	$0.292579\pi$
$212$	1072.82	0.347554
$213$	0	0
$214$	−3068.61	−0.980214
$215$	4488.97	1.42393
$216$	0	0
$217$	5881.53	1.83993
$218$	131.081	0.0407244
$219$	0	0
$220$	3709.35	1.13675
$221$	414.494	0.126162
$222$	0	0
$223$	4450.20	1.33636	0.668178	−	0.744001i	$-0.267074\pi$
$223$	4450.20	1.33636	0.668178	+	0.744001i	$0.267074\pi$
$224$	−5323.72	−1.58797
$225$	0	0
$226$	−767.564	−0.225918
$227$	−5374.73	−1.57151	−0.785756	−	0.618537i	$-0.787726\pi$
$227$	−5374.73	−1.57151	−0.785756	+	0.618537i	$0.787726\pi$
$228$	0	0
$229$	101.013	0.0291490	0.0145745	−	0.999894i	$-0.495361\pi$
$229$	101.013	0.0291490	0.0145745	+	0.999894i	$0.495361\pi$
$230$	−1814.79	−0.520276
$231$	0	0
$232$	5440.57	1.53962
$233$	5256.06	1.47784	0.738918	−	0.673796i	$-0.235337\pi$
$233$	5256.06	1.47784	0.738918	+	0.673796i	$0.235337\pi$
$234$	0	0
$235$	−2031.92	−0.564034
$236$	878.684	0.242362
$237$	0	0
$238$	−712.812	−0.194138
$239$	2571.15	0.695874	0.347937	−	0.937518i	$-0.386882\pi$
$239$	2571.15	0.695874	0.347937	+	0.937518i	$0.386882\pi$
$240$	0	0
$241$	−6085.74	−1.62663	−0.813313	−	0.581826i	$-0.802339\pi$
$241$	−6085.74	−1.62663	−0.813313	+	0.581826i	$0.802339\pi$
$242$	2702.38	0.717834
$243$	0	0
$244$	−343.390	−0.0900956
$245$	5251.08	1.36930
$246$	0	0
$247$	−829.480	−0.213678
$248$	4227.53	1.08245
$249$	0	0
$250$	−2033.53	−0.514447
$251$	−6890.40	−1.73274	−0.866371	−	0.499400i	$-0.833554\pi$
$251$	−6890.40	−1.73274	−0.866371	+	0.499400i	$0.833554\pi$
$252$	0	0
$253$	6093.52	1.51421
$254$	−707.917	−0.174877
$255$	0	0
$256$	−3062.47	−0.747674
$257$	3136.70	0.761332	0.380666	−	0.924713i	$-0.375695\pi$
$257$	3136.70	0.761332	0.380666	+	0.924713i	$0.375695\pi$
$258$	0	0
$259$	−10792.8	−2.58930
$260$	−1608.65	−0.383710
$261$	0	0
$262$	2267.12	0.534592
$263$	−1208.72	−0.283396	−0.141698	−	0.989910i	$-0.545256\pi$
$263$	−1208.72	−0.283396	−0.141698	+	0.989910i	$0.545256\pi$
$264$	0	0
$265$	−2090.32	−0.484556
$266$	1426.47	0.328806
$267$	0	0
$268$	330.329	0.0752913
$269$	810.352	0.183673	0.0918365	−	0.995774i	$-0.470726\pi$
$269$	810.352	0.183673	0.0918365	+	0.995774i	$0.470726\pi$
$270$	0	0
$271$	3795.67	0.850814	0.425407	−	0.905002i	$-0.360131\pi$
$271$	3795.67	0.850814	0.425407	+	0.905002i	$0.360131\pi$
$272$	279.037	0.0622026
$273$	0	0
$274$	−967.454	−0.213307
$275$	−199.729	−0.0437968
$276$	0	0
$277$	−8393.06	−1.82054	−0.910271	−	0.414013i	$-0.864127\pi$
$277$	−8393.06	−1.82054	−0.910271	+	0.414013i	$0.864127\pi$
$278$	−3750.10	−0.809050
$279$	0	0
$280$	6569.69	1.40219
$281$	7161.30	1.52031	0.760156	−	0.649741i	$-0.225123\pi$
$281$	7161.30	1.52031	0.760156	+	0.649741i	$0.225123\pi$
$282$	0	0
$283$	4911.21	1.03159	0.515797	−	0.856711i	$-0.327496\pi$
$283$	4911.21	1.03159	0.515797	+	0.856711i	$0.327496\pi$
$284$	−1542.13	−0.322214
$285$	0	0
$286$	−2024.40	−0.418550
$287$	−12826.0	−2.63795
$288$	0	0
$289$	289.000	0.0588235
$290$	−4463.81	−0.903877
$291$	0	0
$292$	4100.69	0.821831
$293$	−1476.30	−0.294355	−0.147178	−	0.989110i	$-0.547019\pi$
$293$	−1476.30	−0.294355	−0.147178	+	0.989110i	$0.547019\pi$
$294$	0	0
$295$	−1712.06	−0.337899
$296$	−7757.62	−1.52332
$297$	0	0
$298$	−1940.61	−0.377236
$299$	−2642.61	−0.511124
$300$	0	0
$301$	−11241.1	−2.15259
$302$	2837.42	0.540646
$303$	0	0
$304$	−558.405	−0.105351
$305$	669.076	0.125610
$306$	0	0
$307$	2489.64	0.462839	0.231419	−	0.972854i	$-0.425663\pi$
$307$	2489.64	0.462839	0.231419	+	0.972854i	$0.425663\pi$
$308$	−9288.85	−1.71845
$309$	0	0
$310$	−3468.55	−0.635486
$311$	−2546.02	−0.464217	−0.232109	−	0.972690i	$-0.574562\pi$
$311$	−2546.02	−0.464217	−0.232109	+	0.972690i	$0.574562\pi$
$312$	0	0
$313$	4425.28	0.799143	0.399571	−	0.916702i	$-0.369159\pi$
$313$	4425.28	0.799143	0.399571	+	0.916702i	$0.369159\pi$
$314$	−33.0458	−0.00593912
$315$	0	0
$316$	−2964.14	−0.527678
$317$	−1345.29	−0.238356	−0.119178	−	0.992873i	$-0.538026\pi$
$317$	−1345.29	−0.238356	−0.119178	+	0.992873i	$0.538026\pi$
$318$	0	0
$319$	14988.2	2.63065
$320$	1650.76	0.288377
$321$	0	0
$322$	4544.53	0.786512
$323$	−578.342	−0.0996280
$324$	0	0
$325$	86.6177	0.0147836
$326$	2902.87	0.493175
$327$	0	0
$328$	−9219.04	−1.55194
$329$	5088.28	0.852662
$330$	0	0
$331$	4919.61	0.816937	0.408469	−	0.912772i	$-0.366063\pi$
$331$	4919.61	0.816937	0.408469	+	0.912772i	$0.366063\pi$
$332$	−710.536	−0.117457
$333$	0	0
$334$	−2041.20	−0.334399
$335$	−643.627	−0.104971
$336$	0	0
$337$	1151.33	0.186104	0.0930520	−	0.995661i	$-0.470338\pi$
$337$	1151.33	0.186104	0.0930520	+	0.995661i	$0.470338\pi$
$338$	−2366.60	−0.380846
$339$	0	0
$340$	−1121.61	−0.178906
$341$	11646.4	1.84952
$342$	0	0
$343$	−3410.97	−0.536953
$344$	−8079.91	−1.26639
$345$	0	0
$346$	−50.3502	−0.00782325
$347$	2488.64	0.385006	0.192503	−	0.981296i	$-0.438339\pi$
$347$	2488.64	0.385006	0.192503	+	0.981296i	$0.438339\pi$
$348$	0	0
$349$	9301.82	1.42669	0.713345	−	0.700813i	$-0.247179\pi$
$349$	9301.82	1.42669	0.713345	+	0.700813i	$0.247179\pi$
$350$	−148.958	−0.0227489
$351$	0	0
$352$	−10541.8	−1.59625
$353$	3382.00	0.509931	0.254966	−	0.966950i	$-0.417936\pi$
$353$	3382.00	0.509931	0.254966	+	0.966950i	$0.417936\pi$
$354$	0	0
$355$	3004.75	0.449227
$356$	4136.28	0.615793
$357$	0	0
$358$	1672.38	0.246895
$359$	−5846.68	−0.859543	−0.429772	−	0.902938i	$-0.641406\pi$
$359$	−5846.68	−0.859543	−0.429772	+	0.902938i	$0.641406\pi$
$360$	0	0
$361$	−5701.63	−0.831263
$362$	−1763.64	−0.256064
$363$	0	0
$364$	4028.35	0.580062
$365$	−7989.95	−1.14579
$366$	0	0
$367$	−6361.98	−0.904884	−0.452442	−	0.891794i	$-0.649447\pi$
$367$	−6361.98	−0.904884	−0.452442	+	0.891794i	$0.649447\pi$
$368$	−1779.00	−0.252002
$369$	0	0
$370$	6364.88	0.894309
$371$	5234.52	0.732514
$372$	0	0
$373$	−3787.98	−0.525828	−0.262914	−	0.964819i	$-0.584684\pi$
$373$	−3787.98	−0.525828	−0.262914	+	0.964819i	$0.584684\pi$
$374$	−1411.48	−0.195150
$375$	0	0
$376$	3657.35	0.501632
$377$	−6500.00	−0.887977
$378$	0	0
$379$	−11347.7	−1.53797	−0.768987	−	0.639264i	$-0.779239\pi$
$379$	−11347.7	−1.53797	−0.768987	+	0.639264i	$0.779239\pi$
$380$	2244.55	0.303008
$381$	0	0
$382$	−486.439	−0.0651528
$383$	−7001.57	−0.934108	−0.467054	−	0.884229i	$-0.654685\pi$
$383$	−7001.57	−0.934108	−0.467054	+	0.884229i	$0.654685\pi$
$384$	0	0
$385$	18098.8	2.39584
$386$	−3588.78	−0.473224
$387$	0	0
$388$	4854.71	0.635207
$389$	9705.24	1.26498	0.632488	−	0.774570i	$-0.282034\pi$
$389$	9705.24	1.26498	0.632488	+	0.774570i	$0.282034\pi$
$390$	0	0
$391$	−1842.52	−0.238313
$392$	−9451.68	−1.21781
$393$	0	0
$394$	−3474.05	−0.444214
$395$	5775.46	0.735683
$396$	0	0
$397$	−5642.82	−0.713363	−0.356682	−	0.934226i	$-0.616092\pi$
$397$	−5642.82	−0.713363	−0.356682	+	0.934226i	$0.616092\pi$
$398$	−2056.85	−0.259047
$399$	0	0
$400$	58.3110	0.00728887
$401$	6734.68	0.838689	0.419344	−	0.907827i	$-0.362260\pi$
$401$	6734.68	0.838689	0.419344	+	0.907827i	$0.362260\pi$
$402$	0	0
$403$	−5050.75	−0.624307
$404$	9356.22	1.15220
$405$	0	0
$406$	11178.2	1.36641
$407$	−21371.4	−2.60280
$408$	0	0
$409$	15672.4	1.89475	0.947374	−	0.320128i	$-0.103726\pi$
$409$	15672.4	1.89475	0.947374	+	0.320128i	$0.103726\pi$
$410$	7563.93	0.911112
$411$	0	0
$412$	−9200.42	−1.10018
$413$	4287.30	0.510809
$414$	0	0
$415$	1384.44	0.163758
$416$	4571.73	0.538816
$417$	0	0
$418$	2824.64	0.330520
$419$	−4868.91	−0.567690	−0.283845	−	0.958870i	$-0.591610\pi$
$419$	−4868.91	−0.567690	−0.283845	+	0.958870i	$0.591610\pi$
$420$	0	0
$421$	−10710.6	−1.23991	−0.619953	−	0.784639i	$-0.712848\pi$
$421$	−10710.6	−1.23991	−0.619953	+	0.784639i	$0.712848\pi$
$422$	5490.29	0.633325
$423$	0	0
$424$	3762.47	0.430947
$425$	60.3929	0.00689291
$426$	0	0
$427$	−1675.48	−0.189888
$428$	12091.3	1.36555
$429$	0	0
$430$	6629.31	0.743473
$431$	13335.4	1.49036	0.745180	−	0.666864i	$-0.232364\pi$
$431$	13335.4	1.49036	0.745180	+	0.666864i	$0.232364\pi$
$432$	0	0
$433$	11402.7	1.26554	0.632770	−	0.774340i	$-0.281918\pi$
$433$	11402.7	1.26554	0.632770	+	0.774340i	$0.281918\pi$
$434$	8685.85	0.960678
$435$	0	0
$436$	−516.500	−0.0567336
$437$	3687.22	0.403624
$438$	0	0
$439$	−4743.62	−0.515719	−0.257860	−	0.966182i	$-0.583017\pi$
$439$	−4743.62	−0.515719	−0.257860	+	0.966182i	$0.583017\pi$
$440$	13009.0	1.40950
$441$	0	0
$442$	612.125	0.0658729
$443$	−8628.54	−0.925405	−0.462702	−	0.886514i	$-0.653120\pi$
$443$	−8628.54	−0.925405	−0.462702	+	0.886514i	$0.653120\pi$
$444$	0	0
$445$	−8059.30	−0.858534
$446$	6572.06	0.697749
$447$	0	0
$448$	−4133.80	−0.435945
$449$	8566.47	0.900394	0.450197	−	0.892929i	$-0.351354\pi$
$449$	8566.47	0.900394	0.450197	+	0.892929i	$0.351354\pi$
$450$	0	0
$451$	−25397.5	−2.65171
$452$	3024.44	0.314730
$453$	0	0
$454$	−7937.40	−0.820530
$455$	−7849.00	−0.808718
$456$	0	0
$457$	−2971.57	−0.304167	−0.152083	−	0.988368i	$-0.548598\pi$
$457$	−2971.57	−0.304167	−0.152083	+	0.988368i	$0.548598\pi$
$458$	149.176	0.0152195
$459$	0	0
$460$	7150.83	0.724802
$461$	13661.1	1.38018	0.690089	−	0.723724i	$-0.257571\pi$
$461$	13661.1	1.38018	0.690089	+	0.723724i	$0.257571\pi$
$462$	0	0
$463$	−457.083	−0.0458800	−0.0229400	−	0.999737i	$-0.507303\pi$
$463$	−457.083	−0.0458800	−0.0229400	+	0.999737i	$0.507303\pi$
$464$	−4375.80	−0.437804
$465$	0	0
$466$	7762.15	0.771619
$467$	13965.2	1.38379	0.691896	−	0.721997i	$-0.256776\pi$
$467$	13965.2	1.38379	0.691896	+	0.721997i	$0.256776\pi$
$468$	0	0
$469$	1611.75	0.158686
$470$	−3000.74	−0.294498
$471$	0	0
$472$	3081.63	0.300516
$473$	−22259.3	−2.16381
$474$	0	0
$475$	−120.857	−0.0116743
$476$	2808.70	0.270455
$477$	0	0
$478$	3797.08	0.363335
$479$	13827.2	1.31896	0.659481	−	0.751721i	$-0.270776\pi$
$479$	13827.2	1.31896	0.659481	+	0.751721i	$0.270776\pi$
$480$	0	0
$481$	9268.25	0.878578
$482$	−8987.43	−0.849307
$483$	0	0
$484$	−10648.3	−1.00002
$485$	−9459.11	−0.885600
$486$	0	0
$487$	−616.131	−0.0573297	−0.0286648	−	0.999589i	$-0.509126\pi$
$487$	−616.131	−0.0573297	−0.0286648	+	0.999589i	$0.509126\pi$
$488$	−1204.30	−0.111714
$489$	0	0
$490$	7754.80	0.714951
$491$	3845.66	0.353467	0.176733	−	0.984259i	$-0.443447\pi$
$491$	3845.66	0.353467	0.176733	+	0.984259i	$0.443447\pi$
$492$	0	0
$493$	−4532.03	−0.414021
$494$	−1224.98	−0.111567
$495$	0	0
$496$	−3400.16	−0.307806
$497$	−7524.41	−0.679107
$498$	0	0
$499$	−12260.2	−1.09988	−0.549942	−	0.835203i	$-0.685350\pi$
$499$	−12260.2	−1.09988	−0.549942	+	0.835203i	$0.685350\pi$
$500$	8012.75	0.716682
$501$	0	0
$502$	−10175.7	−0.904713
$503$	−15501.4	−1.37410	−0.687050	−	0.726610i	$-0.741095\pi$
$503$	−15501.4	−1.37410	−0.687050	+	0.726610i	$0.741095\pi$
$504$	0	0
$505$	−18230.1	−1.60639
$506$	8998.91	0.790613
$507$	0	0
$508$	2789.42	0.243623
$509$	−1728.98	−0.150562	−0.0752808	−	0.997162i	$-0.523985\pi$
$509$	−1728.98	−0.150562	−0.0752808	+	0.997162i	$0.523985\pi$
$510$	0	0
$511$	20008.2	1.73212
$512$	5757.49	0.496968
$513$	0	0
$514$	4632.29	0.397512
$515$	17926.5	1.53385
$516$	0	0
$517$	10075.6	0.857108
$518$	−15938.7	−1.35195
$519$	0	0
$520$	−5641.70	−0.475779
$521$	−4269.42	−0.359015	−0.179507	−	0.983757i	$-0.557450\pi$
$521$	−4269.42	−0.359015	−0.179507	+	0.983757i	$0.557450\pi$
$522$	0	0
$523$	16223.6	1.35642	0.678211	−	0.734867i	$-0.262755\pi$
$523$	16223.6	1.35642	0.678211	+	0.734867i	$0.262755\pi$
$524$	−8933.17	−0.744747
$525$	0	0
$526$	−1785.04	−0.147969
$527$	−3521.56	−0.291085
$528$	0	0
$529$	−420.016	−0.0345209
$530$	−3086.99	−0.253000
$531$	0	0
$532$	−5620.74	−0.458064
$533$	11014.3	0.895085
$534$	0	0
$535$	−23559.2	−1.90384
$536$	1158.50	0.0933571
$537$	0	0
$538$	1196.73	0.0959008
$539$	−26038.3	−2.08080
$540$	0	0
$541$	24052.0	1.91142	0.955708	−	0.294316i	$-0.0950918\pi$
$541$	24052.0	1.91142	0.955708	+	0.294316i	$0.0950918\pi$
$542$	5605.45	0.444234
$543$	0	0
$544$	3187.57	0.251224
$545$	1006.37	0.0790976
$546$	0	0
$547$	11238.4	0.878467	0.439233	−	0.898373i	$-0.355250\pi$
$547$	11238.4	0.878467	0.439233	+	0.898373i	$0.355250\pi$
$548$	3812.08	0.297160
$549$	0	0
$550$	−294.960	−0.0228675
$551$	9069.43	0.701217
$552$	0	0
$553$	−14462.7	−1.11215
$554$	−12394.9	−0.950556
$555$	0	0
$556$	14776.6	1.12710
$557$	−9466.87	−0.720151	−0.360075	−	0.932923i	$-0.617249\pi$
$557$	−9466.87	−0.720151	−0.360075	+	0.932923i	$0.617249\pi$
$558$	0	0
$559$	9653.30	0.730395
$560$	−5283.94	−0.398727
$561$	0	0
$562$	10575.8	0.793797
$563$	−11403.2	−0.853618	−0.426809	−	0.904342i	$-0.640362\pi$
$563$	−11403.2	−0.853618	−0.426809	+	0.904342i	$0.640362\pi$
$564$	0	0
$565$	−5892.95	−0.438793
$566$	7252.88	0.538624
$567$	0	0
$568$	−5408.40	−0.399527
$569$	−12628.5	−0.930429	−0.465215	−	0.885198i	$-0.654023\pi$
$569$	−12628.5	−0.930429	−0.465215	+	0.885198i	$0.654023\pi$
$570$	0	0
$571$	19907.1	1.45899	0.729497	−	0.683984i	$-0.239754\pi$
$571$	19907.1	1.45899	0.729497	+	0.683984i	$0.239754\pi$
$572$	7976.78	0.583087
$573$	0	0
$574$	−18941.4	−1.37735
$575$	−385.035	−0.0279253
$576$	0	0
$577$	−10632.5	−0.767135	−0.383567	−	0.923513i	$-0.625305\pi$
$577$	−10632.5	−0.767135	−0.383567	+	0.923513i	$0.625305\pi$
$578$	426.795	0.0307134
$579$	0	0
$580$	17588.8	1.25920
$581$	−3466.87	−0.247556
$582$	0	0
$583$	10365.2	0.736334
$584$	14381.5	1.01903
$585$	0	0
$586$	−2180.20	−0.153691
$587$	−24373.6	−1.71381	−0.856905	−	0.515474i	$-0.827616\pi$
$587$	−24373.6	−1.71381	−0.856905	+	0.515474i	$0.827616\pi$
$588$	0	0
$589$	7047.29	0.493003
$590$	−2528.38	−0.176426
$591$	0	0
$592$	6239.38	0.433170
$593$	−21444.6	−1.48503	−0.742517	−	0.669828i	$-0.766368\pi$
$593$	−21444.6	−1.48503	−0.742517	+	0.669828i	$0.766368\pi$
$594$	0	0
$595$	−5472.60	−0.377067
$596$	7646.61	0.525532
$597$	0	0
$598$	−3902.61	−0.266872
$599$	−13133.2	−0.895838	−0.447919	−	0.894074i	$-0.647835\pi$
$599$	−13133.2	−0.895838	−0.447919	+	0.894074i	$0.647835\pi$
$600$	0	0
$601$	8618.09	0.584924	0.292462	−	0.956277i	$-0.405526\pi$
$601$	8618.09	0.584924	0.292462	+	0.956277i	$0.405526\pi$
$602$	−16600.9	−1.12392
$603$	0	0
$604$	−11180.3	−0.753181
$605$	20747.5	1.39422
$606$	0	0
$607$	−14744.9	−0.985956	−0.492978	−	0.870042i	$-0.664092\pi$
$607$	−14744.9	−0.985956	−0.492978	+	0.870042i	$0.664092\pi$
$608$	−6378.92	−0.425492
$609$	0	0
$610$	988.092	0.0655847
$611$	−4369.55	−0.289317
$612$	0	0
$613$	8793.20	0.579371	0.289685	−	0.957122i	$-0.406449\pi$
$613$	8793.20	0.579371	0.289685	+	0.957122i	$0.406449\pi$
$614$	3676.71	0.241661
$615$	0	0
$616$	−32576.9	−2.13078
$617$	−26305.5	−1.71640	−0.858199	−	0.513317i	$-0.828417\pi$
$617$	−26305.5	−1.71640	−0.858199	+	0.513317i	$0.828417\pi$
$618$	0	0
$619$	14910.3	0.968170	0.484085	−	0.875021i	$-0.339153\pi$
$619$	14910.3	0.968170	0.484085	+	0.875021i	$0.339153\pi$
$620$	13667.2	0.885303
$621$	0	0
$622$	−3759.96	−0.242381
$623$	20181.9	1.29786
$624$	0	0
$625$	−16056.4	−1.02761
$626$	6535.26	0.417255
$627$	0	0
$628$	130.211	0.00827386
$629$	6462.15	0.409639
$630$	0	0
$631$	−25270.5	−1.59430	−0.797151	−	0.603780i	$-0.793660\pi$
$631$	−25270.5	−1.59430	−0.797151	+	0.603780i	$0.793660\pi$
$632$	−10395.5	−0.654291
$633$	0	0
$634$	−1986.72	−0.124453
$635$	−5435.01	−0.339657
$636$	0	0
$637$	11292.2	0.702375
$638$	22134.5	1.37353
$639$	0	0
$640$	−14569.7	−0.899870
$641$	21978.2	1.35427	0.677135	−	0.735859i	$-0.263221\pi$
$641$	21978.2	1.35427	0.677135	+	0.735859i	$0.263221\pi$
$642$	0	0
$643$	−15814.1	−0.969902	−0.484951	−	0.874541i	$-0.661163\pi$
$643$	−15814.1	−0.969902	−0.484951	+	0.874541i	$0.661163\pi$
$644$	−17906.9	−1.09570
$645$	0	0
$646$	−854.096	−0.0520185
$647$	−27090.8	−1.64614	−0.823068	−	0.567942i	$-0.807740\pi$
$647$	−27090.8	−1.64614	−0.823068	+	0.567942i	$0.807740\pi$
$648$	0	0
$649$	8489.55	0.513473
$650$	127.917	0.00771895
$651$	0	0
$652$	−11438.2	−0.687048
$653$	−27121.1	−1.62531	−0.812656	−	0.582743i	$-0.801979\pi$
$653$	−27121.1	−1.62531	−0.812656	+	0.582743i	$0.801979\pi$
$654$	0	0
$655$	17405.8	1.03832
$656$	7414.79	0.441309
$657$	0	0
$658$	7514.37	0.445199
$659$	18075.1	1.06845	0.534224	−	0.845343i	$-0.320604\pi$
$659$	18075.1	1.06845	0.534224	+	0.845343i	$0.320604\pi$
$660$	0	0
$661$	−404.297	−0.0237902	−0.0118951	−	0.999929i	$-0.503786\pi$
$661$	−404.297	−0.0237902	−0.0118951	+	0.999929i	$0.503786\pi$
$662$	7265.28	0.426546
$663$	0	0
$664$	−2491.92	−0.145640
$665$	10951.7	0.638628
$666$	0	0
$667$	28894.0	1.67733
$668$	8042.96	0.465855
$669$	0	0
$670$	−950.509	−0.0548080
$671$	−3317.72	−0.190878
$672$	0	0
$673$	−18869.1	−1.08076	−0.540379	−	0.841422i	$-0.681719\pi$
$673$	−18869.1	−1.08076	−0.540379	+	0.841422i	$0.681719\pi$
$674$	1700.29	0.0971700
$675$	0	0
$676$	9325.14	0.530561
$677$	9061.24	0.514404	0.257202	−	0.966358i	$-0.417199\pi$
$677$	9061.24	0.514404	0.257202	+	0.966358i	$0.417199\pi$
$678$	0	0
$679$	23687.2	1.33878
$680$	−3933.59	−0.221833
$681$	0	0
$682$	17199.4	0.965687
$683$	5991.75	0.335678	0.167839	−	0.985814i	$-0.446321\pi$
$683$	5991.75	0.335678	0.167839	+	0.985814i	$0.446321\pi$
$684$	0	0
$685$	−7427.61	−0.414298
$686$	−5037.32	−0.280358
$687$	0	0
$688$	6498.59	0.360111
$689$	−4495.13	−0.248550
$690$	0	0
$691$	20027.9	1.10260	0.551300	−	0.834307i	$-0.314132\pi$
$691$	20027.9	1.10260	0.551300	+	0.834307i	$0.314132\pi$
$692$	198.396	0.0108987
$693$	0	0
$694$	3675.22	0.201022
$695$	−28791.3	−1.57139
$696$	0	0
$697$	7679.53	0.417335
$698$	13736.9	0.744915
$699$	0	0
$700$	586.940	0.0316918
$701$	−3034.01	−0.163471	−0.0817354	−	0.996654i	$-0.526046\pi$
$701$	−3034.01	−0.163471	−0.0817354	+	0.996654i	$0.526046\pi$
$702$	0	0
$703$	−12932.0	−0.693795
$704$	−8185.58	−0.438219
$705$	0	0
$706$	4994.54	0.266249
$707$	45651.2	2.42841
$708$	0	0
$709$	−27012.7	−1.43087	−0.715433	−	0.698681i	$-0.753770\pi$
$709$	−27012.7	−1.43087	−0.715433	+	0.698681i	$0.753770\pi$
$710$	4437.42	0.234554
$711$	0	0
$712$	14506.3	0.763550
$713$	22451.7	1.17928
$714$	0	0
$715$	−15542.3	−0.812935
$716$	−6589.72	−0.343952
$717$	0	0
$718$	−8634.38	−0.448791
$719$	−18613.8	−0.965478	−0.482739	−	0.875764i	$-0.660358\pi$
$719$	−18613.8	−0.965478	−0.482739	+	0.875764i	$0.660358\pi$
$720$	0	0
$721$	−44891.0	−2.31876
$722$	−8420.17	−0.434025
$723$	0	0
$724$	6949.31	0.356725
$725$	−947.068	−0.0485148
$726$	0	0
$727$	8277.72	0.422289	0.211144	−	0.977455i	$-0.432281\pi$
$727$	8277.72	0.422289	0.211144	+	0.977455i	$0.432281\pi$
$728$	14127.8	0.719245
$729$	0	0
$730$	−11799.6	−0.598249
$731$	6730.62	0.340548
$732$	0	0
$733$	−16336.3	−0.823188	−0.411594	−	0.911367i	$-0.635028\pi$
$733$	−16336.3	−0.823188	−0.411594	+	0.911367i	$0.635028\pi$
$734$	−9395.37	−0.472465
$735$	0	0
$736$	−20322.3	−1.01779
$737$	3191.53	0.159514
$738$	0	0
$739$	5768.22	0.287128	0.143564	−	0.989641i	$-0.454144\pi$
$739$	5768.22	0.287128	0.143564	+	0.989641i	$0.454144\pi$
$740$	−25079.6	−1.24587
$741$	0	0
$742$	7730.34	0.382466
$743$	5166.75	0.255114	0.127557	−	0.991831i	$-0.459286\pi$
$743$	5166.75	0.255114	0.127557	+	0.991831i	$0.459286\pi$
$744$	0	0
$745$	−14899.0	−0.732693
$746$	−5594.08	−0.274550
$747$	0	0
$748$	5561.69	0.271866
$749$	58996.2	2.87807
$750$	0	0
$751$	5015.28	0.243689	0.121844	−	0.992549i	$-0.461119\pi$
$751$	5015.28	0.243689	0.121844	+	0.992549i	$0.461119\pi$
$752$	−2941.57	−0.142644
$753$	0	0
$754$	−9599.21	−0.463637
$755$	21784.2	1.05008
$756$	0	0
$757$	26385.2	1.26683	0.633413	−	0.773814i	$-0.281653\pi$
$757$	26385.2	1.26683	0.633413	+	0.773814i	$0.281653\pi$
$758$	−16758.3	−0.803019
$759$	0	0
$760$	7871.85	0.375713
$761$	41519.7	1.97778	0.988889	−	0.148655i	$-0.0474945\pi$
$761$	41519.7	1.97778	0.988889	+	0.148655i	$0.0474945\pi$
$762$	0	0
$763$	−2520.12	−0.119573
$764$	1916.72	0.0907652
$765$	0	0
$766$	−10339.9	−0.487724
$767$	−3681.71	−0.173323
$768$	0	0
$769$	22527.4	1.05639	0.528193	−	0.849124i	$-0.322870\pi$
$769$	22527.4	1.05639	0.528193	+	0.849124i	$0.322870\pi$
$770$	26728.3	1.25094
$771$	0	0
$772$	14140.9	0.659253
$773$	25447.2	1.18405	0.592026	−	0.805919i	$-0.298328\pi$
$773$	25447.2	1.18405	0.592026	+	0.805919i	$0.298328\pi$
$774$	0	0
$775$	−735.908	−0.0341091
$776$	17025.9	0.787622
$777$	0	0
$778$	14332.7	0.660479
$779$	−15368.1	−0.706831
$780$	0	0
$781$	−14899.6	−0.682648
$782$	−2721.03	−0.124430
$783$	0	0
$784$	7601.89	0.346296
$785$	−253.708	−0.0115353
$786$	0	0
$787$	−3417.03	−0.154770	−0.0773849	−	0.997001i	$-0.524657\pi$
$787$	−3417.03	−0.154770	−0.0773849	+	0.997001i	$0.524657\pi$
$788$	13688.9	0.618840
$789$	0	0
$790$	8529.20	0.384121
$791$	14757.0	0.663334
$792$	0	0
$793$	1438.81	0.0644310
$794$	−8333.33	−0.372467
$795$	0	0
$796$	8104.64	0.360881
$797$	10295.8	0.457587	0.228793	−	0.973475i	$-0.426522\pi$
$797$	10295.8	0.457587	0.228793	+	0.973475i	$0.426522\pi$
$798$	0	0
$799$	−3046.60	−0.134895
$800$	666.113	0.0294383
$801$	0	0
$802$	9945.78	0.437903
$803$	39619.5	1.74115
$804$	0	0
$805$	34890.5	1.52761
$806$	−7458.95	−0.325968
$807$	0	0
$808$	32813.2	1.42867
$809$	−35143.9	−1.52731	−0.763654	−	0.645626i	$-0.776597\pi$
$809$	−35143.9	−1.52731	−0.763654	+	0.645626i	$0.776597\pi$
$810$	0	0
$811$	28871.8	1.25009	0.625047	−	0.780587i	$-0.285080\pi$
$811$	28871.8	1.25009	0.625047	+	0.780587i	$0.285080\pi$
$812$	−44045.5	−1.90356
$813$	0	0
$814$	−31561.3	−1.35900
$815$	22286.7	0.957876
$816$	0	0
$817$	−13469.2	−0.576778
$818$	23145.1	0.989301
$819$	0	0
$820$	−29804.3	−1.26928
$821$	−5993.62	−0.254785	−0.127393	−	0.991852i	$-0.540661\pi$
$821$	−5993.62	−0.254785	−0.127393	+	0.991852i	$0.540661\pi$
$822$	0	0
$823$	−15532.0	−0.657853	−0.328927	−	0.944356i	$-0.606687\pi$
$823$	−15532.0	−0.657853	−0.328927	+	0.944356i	$0.606687\pi$
$824$	−32266.7	−1.36416
$825$	0	0
$826$	6331.49	0.266708
$827$	13605.2	0.572068	0.286034	−	0.958219i	$-0.407663\pi$
$827$	13605.2	0.572068	0.286034	+	0.958219i	$0.407663\pi$
$828$	0	0
$829$	7835.02	0.328253	0.164126	−	0.986439i	$-0.447520\pi$
$829$	7835.02	0.328253	0.164126	+	0.986439i	$0.447520\pi$
$830$	2044.54	0.0855024
$831$	0	0
$832$	3549.89	0.147921
$833$	7873.31	0.327484
$834$	0	0
$835$	−15671.2	−0.649491
$836$	−11130.0	−0.460452
$837$	0	0
$838$	−7190.41	−0.296407
$839$	−29621.9	−1.21890	−0.609452	−	0.792823i	$-0.708610\pi$
$839$	−29621.9	−1.21890	−0.609452	+	0.792823i	$0.708610\pi$
$840$	0	0
$841$	46681.3	1.91403
$842$	−15817.4	−0.647389
$843$	0	0
$844$	−21633.5	−0.882293
$845$	−18169.5	−0.739704
$846$	0	0
$847$	−51955.3	−2.10768
$848$	−3026.12	−0.122544
$849$	0	0
$850$	89.1883	0.00359898
$851$	−41199.4	−1.65958
$852$	0	0
$853$	−9476.96	−0.380405	−0.190202	−	0.981745i	$-0.560914\pi$
$853$	−9476.96	−0.380405	−0.190202	+	0.981745i	$0.560914\pi$
$854$	−2474.35	−0.0991458
$855$	0	0
$856$	42405.3	1.69320
$857$	−28763.9	−1.14651	−0.573253	−	0.819378i	$-0.694319\pi$
$857$	−28763.9	−1.14651	−0.573253	+	0.819378i	$0.694319\pi$
$858$	0	0
$859$	17233.0	0.684497	0.342249	−	0.939609i	$-0.388812\pi$
$859$	17233.0	0.684497	0.342249	+	0.939609i	$0.388812\pi$
$860$	−26121.6	−1.03574
$861$	0	0
$862$	19693.8	0.778158
$863$	−28659.4	−1.13045	−0.565225	−	0.824937i	$-0.691211\pi$
$863$	−28659.4	−1.13045	−0.565225	+	0.824937i	$0.691211\pi$
$864$	0	0
$865$	−386.562	−0.0151948
$866$	16839.5	0.660773
$867$	0	0
$868$	−34225.0	−1.33833
$869$	−28638.6	−1.11795
$870$	0	0
$871$	−1384.09	−0.0538439
$872$	−1811.41	−0.0703466
$873$	0	0
$874$	5445.29	0.210743
$875$	39096.1	1.51050
$876$	0	0
$877$	3227.71	0.124278	0.0621391	−	0.998067i	$-0.480208\pi$
$877$	3227.71	0.124278	0.0621391	+	0.998067i	$0.480208\pi$
$878$	−7005.38	−0.269271
$879$	0	0
$880$	−10463.0	−0.400806
$881$	−27737.4	−1.06072	−0.530361	−	0.847772i	$-0.677944\pi$
$881$	−27737.4	−1.06072	−0.530361	+	0.847772i	$0.677944\pi$
$882$	0	0
$883$	4939.63	0.188258	0.0941290	−	0.995560i	$-0.469993\pi$
$883$	4939.63	0.188258	0.0941290	+	0.995560i	$0.469993\pi$
$884$	−2411.97	−0.0917684
$885$	0	0
$886$	−12742.6	−0.483179
$887$	28594.1	1.08241	0.541204	−	0.840892i	$-0.317969\pi$
$887$	28594.1	1.08241	0.541204	+	0.840892i	$0.317969\pi$
$888$	0	0
$889$	13610.2	0.513466
$890$	−11902.0	−0.448264
$891$	0	0
$892$	−25896.0	−0.972042
$893$	6096.81	0.228468
$894$	0	0
$895$	12839.7	0.479535
$896$	36484.9	1.36035
$897$	0	0
$898$	12651.0	0.470121
$899$	55224.3	2.04876
$900$	0	0
$901$	−3134.16	−0.115887
$902$	−37507.0	−1.38453
$903$	0	0
$904$	10607.0	0.390248
$905$	−13540.3	−0.497343
$906$	0	0
$907$	−24054.7	−0.880623	−0.440311	−	0.897845i	$-0.645132\pi$
$907$	−24054.7	−0.880623	−0.440311	+	0.897845i	$0.645132\pi$
$908$	31275.9	1.14309
$909$	0	0
$910$	−11591.4	−0.422254
$911$	3489.44	0.126905	0.0634525	−	0.997985i	$-0.479789\pi$
$911$	3489.44	0.126905	0.0634525	+	0.997985i	$0.479789\pi$
$912$	0	0
$913$	−6864.96	−0.248847
$914$	−4388.42	−0.158814
$915$	0	0
$916$	−587.800	−0.0212025
$917$	−43587.0	−1.56965
$918$	0	0
$919$	−42107.0	−1.51141	−0.755703	−	0.654915i	$-0.772704\pi$
$919$	−42107.0	−1.51141	−0.755703	+	0.654915i	$0.772704\pi$
$920$	25078.6	0.898715
$921$	0	0
$922$	20174.8	0.720630
$923$	6461.57	0.230428
$924$	0	0
$925$	1350.41	0.0480012
$926$	−675.020	−0.0239552
$927$	0	0
$928$	−49986.7	−1.76821
$929$	−20282.1	−0.716289	−0.358145	−	0.933666i	$-0.616591\pi$
$929$	−20282.1	−0.716289	−0.358145	+	0.933666i	$0.616591\pi$
$930$	0	0
$931$	−15756.0	−0.554651
$932$	−30585.3	−1.07495
$933$	0	0
$934$	20623.8	0.722516
$935$	−10836.6	−0.379033
$936$	0	0
$937$	25524.5	0.889912	0.444956	−	0.895552i	$-0.353219\pi$
$937$	25524.5	0.889912	0.444956	+	0.895552i	$0.353219\pi$
$938$	2380.24	0.0828545
$939$	0	0
$940$	11823.9	0.410268
$941$	7699.76	0.266743	0.133371	−	0.991066i	$-0.457420\pi$
$941$	7699.76	0.266743	0.133371	+	0.991066i	$0.457420\pi$
$942$	0	0
$943$	−48960.8	−1.69076
$944$	−2478.52	−0.0854545
$945$	0	0
$946$	−32872.5	−1.12979
$947$	8302.26	0.284886	0.142443	−	0.989803i	$-0.454504\pi$
$947$	8302.26	0.284886	0.142443	+	0.989803i	$0.454504\pi$
$948$	0	0
$949$	−17182.0	−0.587725
$950$	−178.482	−0.00609550
$951$	0	0
$952$	9850.39	0.335350
$953$	−12381.3	−0.420850	−0.210425	−	0.977610i	$-0.567485\pi$
$953$	−12381.3	−0.420850	−0.210425	+	0.977610i	$0.567485\pi$
$954$	0	0
$955$	−3734.62	−0.126544
$956$	−14961.7	−0.506167
$957$	0	0
$958$	20420.1	0.688667
$959$	18600.0	0.626304
$960$	0	0
$961$	13120.4	0.440415
$962$	13687.4	0.458730
$963$	0	0
$964$	35413.3	1.18318
$965$	−27552.8	−0.919125
$966$	0	0
$967$	−28463.8	−0.946573	−0.473286	−	0.880909i	$-0.656932\pi$
$967$	−28463.8	−0.946573	−0.473286	+	0.880909i	$0.656932\pi$
$968$	−37344.4	−1.23997
$969$	0	0
$970$	−13969.2	−0.462397
$971$	6883.28	0.227492	0.113746	−	0.993510i	$-0.463715\pi$
$971$	6883.28	0.227492	0.113746	+	0.993510i	$0.463715\pi$
$972$	0	0
$973$	72098.3	2.37550
$974$	−909.902	−0.0299334
$975$	0	0
$976$	968.609	0.0317668
$977$	10994.4	0.360023	0.180011	−	0.983665i	$-0.442387\pi$
$977$	10994.4	0.360023	0.180011	+	0.983665i	$0.442387\pi$
$978$	0	0
$979$	39963.3	1.30463
$980$	−30556.4	−0.996008
$981$	0	0
$982$	5679.27	0.184555
$983$	17130.4	0.555822	0.277911	−	0.960607i	$-0.410358\pi$
$983$	17130.4	0.555822	0.277911	+	0.960607i	$0.410358\pi$
$984$	0	0
$985$	−26672.0	−0.862781
$986$	−6692.91	−0.216172
$987$	0	0
$988$	4826.79	0.155426
$989$	−42911.1	−1.37967
$990$	0	0
$991$	40455.8	1.29679	0.648396	−	0.761304i	$-0.275440\pi$
$991$	40455.8	1.29679	0.648396	+	0.761304i	$0.275440\pi$
$992$	−38841.6	−1.24317
$993$	0	0
$994$	−11112.1	−0.354581
$995$	−15791.4	−0.503138
$996$	0	0
$997$	−61277.0	−1.94650	−0.973251	−	0.229746i	$-0.926210\pi$
$997$	−61277.0	−1.94650	−0.973251	+	0.229746i	$0.926210\pi$
$998$	−18105.9	−0.574280
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.a.h.1.3	✓	4
3.2	odd	2		153.4.a.i.1.2	yes	4
4.3	odd	2		2448.4.a.bo.1.4		4
12.11	even	2		2448.4.a.bs.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.a.h.1.3	✓	4	1.1	even	1	trivial
153.4.a.i.1.2	yes	4	3.2	odd	2
2448.4.a.bo.1.4		4	4.3	odd	2
2448.4.a.bs.1.1		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	153.a (trivial)

\(f(q)\)	\(=\)	\(q+1.47680 q^{2} -5.81906 q^{4} +11.3381 q^{5} -28.3925 q^{7} -20.4080 q^{8} +O(q^{10})\)	\(q+1.47680 q^{2} -5.81906 q^{4} +11.3381 q^{5} -28.3925 q^{7} -20.4080 q^{8} +16.7441 q^{10} -56.2218 q^{11} +24.3820 q^{13} -41.9301 q^{14} +16.4140 q^{16} +17.0000 q^{17} -34.0201 q^{19} -65.9771 q^{20} -83.0284 q^{22} -108.384 q^{23} +3.55252 q^{25} +36.0074 q^{26} +165.218 q^{28} -266.590 q^{29} -207.151 q^{31} +187.504 q^{32} +25.1056 q^{34} -321.917 q^{35} +380.126 q^{37} -50.2410 q^{38} -231.388 q^{40} +451.737 q^{41} +395.919 q^{43} +327.158 q^{44} -160.061 q^{46} -179.212 q^{47} +463.136 q^{49} +5.24637 q^{50} -141.880 q^{52} -184.363 q^{53} -637.449 q^{55} +579.435 q^{56} -393.700 q^{58} -151.001 q^{59} +59.0113 q^{61} -305.920 q^{62} +145.594 q^{64} +276.446 q^{65} -56.7668 q^{67} -98.9240 q^{68} -475.408 q^{70} +265.014 q^{71} -704.699 q^{73} +561.371 q^{74} +197.965 q^{76} +1596.28 q^{77} +509.385 q^{79} +186.103 q^{80} +667.125 q^{82} +122.105 q^{83} +192.748 q^{85} +584.693 q^{86} +1147.37 q^{88} -710.816 q^{89} -692.267 q^{91} +630.690 q^{92} -264.660 q^{94} -385.724 q^{95} -834.277 q^{97} +683.960 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8	\( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8} - 2 q^{10} - 50 q^{11} + 26 q^{13} - 80 q^{14} + 138 q^{16} + 68 q^{17} + 34 q^{19} - 312 q^{20} - 254 q^{22} - 382 q^{23} + 138 q^{25} + 22 q^{26} + 52 q^{28} - 540 q^{29} - 356 q^{31} - 730 q^{32} - 68 q^{34} - 304 q^{35} - 404 q^{37} - 298 q^{38} + 332 q^{40} + 114 q^{41} + 570 q^{43} + 1368 q^{44} - 290 q^{46} - 496 q^{47} - 224 q^{49} + 1862 q^{50} - 1012 q^{52} - 92 q^{53} - 482 q^{55} + 1428 q^{56} + 1324 q^{58} + 48 q^{59} - 1036 q^{61} + 2564 q^{62} + 2898 q^{64} + 342 q^{65} + 812 q^{67} + 442 q^{68} + 152 q^{70} - 1044 q^{71} - 1212 q^{73} + 1444 q^{74} + 2268 q^{76} + 564 q^{77} + 488 q^{79} + 1000 q^{80} - 938 q^{82} - 1708 q^{83} - 374 q^{85} + 2446 q^{86} - 3868 q^{88} + 8 q^{89} + 716 q^{91} - 1356 q^{92} - 1224 q^{94} - 1010 q^{95} - 76 q^{97} + 1472 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8 - 2 * q^10 - 50 * q^11 + 26 * q^13 - 80 * q^14 + 138 * q^16 + 68 * q^17 + 34 * q^19 - 312 * q^20 - 254 * q^22 - 382 * q^23 + 138 * q^25 + 22 * q^26 + 52 * q^28 - 540 * q^29 - 356 * q^31 - 730 * q^32 - 68 * q^34 - 304 * q^35 - 404 * q^37 - 298 * q^38 + 332 * q^40 + 114 * q^41 + 570 * q^43 + 1368 * q^44 - 290 * q^46 - 496 * q^47 - 224 * q^49 + 1862 * q^50 - 1012 * q^52 - 92 * q^53 - 482 * q^55 + 1428 * q^56 + 1324 * q^58 + 48 * q^59 - 1036 * q^61 + 2564 * q^62 + 2898 * q^64 + 342 * q^65 + 812 * q^67 + 442 * q^68 + 152 * q^70 - 1044 * q^71 - 1212 * q^73 + 1444 * q^74 + 2268 * q^76 + 564 * q^77 + 488 * q^79 + 1000 * q^80 - 938 * q^82 - 1708 * q^83 - 374 * q^85 + 2446 * q^86 - 3868 * q^88 + 8 * q^89 + 716 * q^91 - 1356 * q^92 - 1224 * q^94 - 1010 * q^95 - 76 * q^97 + 1472 * q^98

Introduction

L-functions

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Database

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