LMFDB - Embedded newform 531.4.a.c.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,4,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, 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("531.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	531.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:	$31.3300142130$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 41x^{5} - 7x^{4} + 484x^{3} + 63x^{2} - 1736x - 44 $ x^7 - 41x^5 - 7x^4 + 484x^3 + 63x^2 - 1736*x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.61892$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61892 q^{2} -1.14126 q^{4} +7.96684 q^{5} +16.8751 q^{7} -23.9402 q^{8} +O(q^{10})$	\(q+2.61892 q^{2} -1.14126 q^{4} +7.96684 q^{5} +16.8751 q^{7} -23.9402 q^{8} +20.8645 q^{10} -62.7682 q^{11} -75.2191 q^{13} +44.1946 q^{14} -53.5675 q^{16} +51.9804 q^{17} +24.2087 q^{19} -9.09222 q^{20} -164.385 q^{22} +9.85642 q^{23} -61.5295 q^{25} -196.993 q^{26} -19.2589 q^{28} -160.344 q^{29} -125.861 q^{31} +51.2329 q^{32} +136.132 q^{34} +134.441 q^{35} -224.138 q^{37} +63.4006 q^{38} -190.728 q^{40} +467.472 q^{41} +90.9608 q^{43} +71.6348 q^{44} +25.8132 q^{46} -437.881 q^{47} -58.2301 q^{49} -161.141 q^{50} +85.8445 q^{52} +353.688 q^{53} -500.064 q^{55} -403.994 q^{56} -419.929 q^{58} -59.0000 q^{59} -697.325 q^{61} -329.620 q^{62} +562.715 q^{64} -599.258 q^{65} -292.026 q^{67} -59.3231 q^{68} +352.091 q^{70} -223.496 q^{71} +122.800 q^{73} -586.998 q^{74} -27.6284 q^{76} -1059.22 q^{77} -440.541 q^{79} -426.763 q^{80} +1224.27 q^{82} +662.365 q^{83} +414.119 q^{85} +238.219 q^{86} +1502.68 q^{88} -1012.63 q^{89} -1269.33 q^{91} -11.2487 q^{92} -1146.77 q^{94} +192.867 q^{95} -397.912 q^{97} -152.500 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8}+O(q^{10}) $ 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8	$ 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8} - 71 q^{10} + 5 q^{11} - 67 q^{13} + 65 q^{14} - 94 q^{16} + 23 q^{17} - 176 q^{19} + 207 q^{20} - 704 q^{22} + 218 q^{23} - 183 q^{25} - 58 q^{26} - 938 q^{28} - 168 q^{29} - 604 q^{31} + 448 q^{32} - 610 q^{34} + 336 q^{35} - 505 q^{37} + 453 q^{38} - 1080 q^{40} + 265 q^{41} - 493 q^{43} - 504 q^{44} + 381 q^{46} + 244 q^{47} + 770 q^{49} - 1639 q^{50} + 160 q^{52} - 686 q^{53} - 116 q^{55} - 2190 q^{56} + 1584 q^{58} - 413 q^{59} - 838 q^{61} - 286 q^{62} + 205 q^{64} - 490 q^{65} - 1504 q^{67} - 3047 q^{68} + 1530 q^{70} + 1267 q^{71} - 666 q^{73} - 528 q^{74} - 64 q^{76} - 1109 q^{77} - 2741 q^{79} - 1213 q^{80} + 953 q^{82} + 2025 q^{83} - 1274 q^{85} - 4394 q^{86} - 1639 q^{88} - 616 q^{89} - 2415 q^{91} - 218 q^{92} + 900 q^{94} - 2554 q^{95} - 1298 q^{97} + 172 q^{98}+O(q^{100}) $ 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8 - 71 * q^10 + 5 * q^11 - 67 * q^13 + 65 * q^14 - 94 * q^16 + 23 * q^17 - 176 * q^19 + 207 * q^20 - 704 * q^22 + 218 * q^23 - 183 * q^25 - 58 * q^26 - 938 * q^28 - 168 * q^29 - 604 * q^31 + 448 * q^32 - 610 * q^34 + 336 * q^35 - 505 * q^37 + 453 * q^38 - 1080 * q^40 + 265 * q^41 - 493 * q^43 - 504 * q^44 + 381 * q^46 + 244 * q^47 + 770 * q^49 - 1639 * q^50 + 160 * q^52 - 686 * q^53 - 116 * q^55 - 2190 * q^56 + 1584 * q^58 - 413 * q^59 - 838 * q^61 - 286 * q^62 + 205 * q^64 - 490 * q^65 - 1504 * q^67 - 3047 * q^68 + 1530 * q^70 + 1267 * q^71 - 666 * q^73 - 528 * q^74 - 64 * q^76 - 1109 * q^77 - 2741 * q^79 - 1213 * q^80 + 953 * q^82 + 2025 * q^83 - 1274 * q^85 - 4394 * q^86 - 1639 * q^88 - 616 * q^89 - 2415 * q^91 - 218 * q^92 + 900 * q^94 - 2554 * q^95 - 1298 * q^97 + 172 * 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 $

$2$	2.61892	0.925928	0.462964	−	0.886377i	$-0.346786\pi$
$2$	2.61892	0.925928	0.462964	+	0.886377i	$0.346786\pi$
$3$	0	0
$3$	0	0
$4$	−1.14126	−0.142657
$5$	7.96684	0.712575	0.356288	−	0.934376i	$-0.384042\pi$
$5$	7.96684	0.712575	0.356288	+	0.934376i	$0.384042\pi$
$6$	0	0
$7$	16.8751	0.911171	0.455586	−	0.890192i	$-0.349430\pi$
$7$	16.8751	0.911171	0.455586	+	0.890192i	$0.349430\pi$
$8$	−23.9402	−1.05802
$9$	0	0
$10$	20.8645	0.659793
$11$	−62.7682	−1.72048	−0.860242	−	0.509886i	$-0.829687\pi$
$11$	−62.7682	−1.72048	−0.860242	+	0.509886i	$0.829687\pi$
$12$	0	0
$13$	−75.2191	−1.60477	−0.802385	−	0.596807i	$-0.796436\pi$
$13$	−75.2191	−1.60477	−0.802385	+	0.596807i	$0.796436\pi$
$14$	44.1946	0.843679
$15$	0	0
$16$	−53.5675	−0.836991
$17$	51.9804	0.741593	0.370797	−	0.928714i	$-0.379085\pi$
$17$	51.9804	0.741593	0.370797	+	0.928714i	$0.379085\pi$
$18$	0	0
$19$	24.2087	0.292308	0.146154	−	0.989262i	$-0.453310\pi$
$19$	24.2087	0.292308	0.146154	+	0.989262i	$0.453310\pi$
$20$	−9.09222	−0.101654
$21$	0	0
$22$	−164.385	−1.59304
$23$	9.85642	0.0893568	0.0446784	−	0.999001i	$-0.485774\pi$
$23$	9.85642	0.0893568	0.0446784	+	0.999001i	$0.485774\pi$
$24$	0	0
$25$	−61.5295	−0.492236
$26$	−196.993	−1.48590
$27$	0	0
$28$	−19.2589	−0.129985
$29$	−160.344	−1.02673	−0.513365	−	0.858170i	$-0.671601\pi$
$29$	−160.344	−1.02673	−0.513365	+	0.858170i	$0.671601\pi$
$30$	0	0
$31$	−125.861	−0.729203	−0.364602	−	0.931164i	$-0.618795\pi$
$31$	−125.861	−0.729203	−0.364602	+	0.931164i	$0.618795\pi$
$32$	51.2329	0.283025
$33$	0	0
$34$	136.132	0.686662
$35$	134.441	0.649278
$36$	0	0
$37$	−224.138	−0.995892	−0.497946	−	0.867208i	$-0.665912\pi$
$37$	−224.138	−0.995892	−0.497946	+	0.867208i	$0.665912\pi$
$38$	63.4006	0.270656
$39$	0	0
$40$	−190.728	−0.753918
$41$	467.472	1.78066	0.890328	−	0.455319i	$-0.150475\pi$
$41$	467.472	1.78066	0.890328	+	0.455319i	$0.150475\pi$
$42$	0	0
$43$	90.9608	0.322591	0.161295	−	0.986906i	$-0.448433\pi$
$43$	90.9608	0.322591	0.161295	+	0.986906i	$0.448433\pi$
$44$	71.6348	0.245440
$45$	0	0
$46$	25.8132	0.0827379
$47$	−437.881	−1.35897	−0.679484	−	0.733691i	$-0.737796\pi$
$47$	−437.881	−1.35897	−0.679484	+	0.733691i	$0.737796\pi$
$48$	0	0
$49$	−58.2301	−0.169767
$50$	−161.141	−0.455775
$51$	0	0
$52$	85.8445	0.228932
$53$	353.688	0.916655	0.458328	−	0.888783i	$-0.348449\pi$
$53$	353.688	0.916655	0.458328	+	0.888783i	$0.348449\pi$
$54$	0	0
$55$	−500.064	−1.22597
$56$	−403.994	−0.964036
$57$	0	0
$58$	−419.929	−0.950678
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	0	0
$61$	−697.325	−1.46366	−0.731831	−	0.681487i	$-0.761334\pi$
$61$	−697.325	−1.46366	−0.731831	+	0.681487i	$0.761334\pi$
$62$	−329.620	−0.675190
$63$	0	0
$64$	562.715	1.09905
$65$	−599.258	−1.14352
$66$	0	0
$67$	−292.026	−0.532487	−0.266244	−	0.963906i	$-0.585783\pi$
$67$	−292.026	−0.532487	−0.266244	+	0.963906i	$0.585783\pi$
$68$	−59.3231	−0.105794
$69$	0	0
$70$	352.091	0.601185
$71$	−223.496	−0.373579	−0.186789	−	0.982400i	$-0.559808\pi$
$71$	−223.496	−0.373579	−0.186789	+	0.982400i	$0.559808\pi$
$72$	0	0
$73$	122.800	0.196885	0.0984426	−	0.995143i	$-0.468614\pi$
$73$	122.800	0.196885	0.0984426	+	0.995143i	$0.468614\pi$
$74$	−586.998	−0.922124
$75$	0	0
$76$	−27.6284	−0.0416999
$77$	−1059.22	−1.56766
$78$	0	0
$79$	−440.541	−0.627401	−0.313701	−	0.949522i	$-0.601569\pi$
$79$	−440.541	−0.627401	−0.313701	+	0.949522i	$0.601569\pi$
$80$	−426.763	−0.596420
$81$	0	0
$82$	1224.27	1.64876
$83$	662.365	0.875951	0.437976	−	0.898987i	$-0.355696\pi$
$83$	662.365	0.875951	0.437976	+	0.898987i	$0.355696\pi$
$84$	0	0
$85$	414.119	0.528441
$86$	238.219	0.298696
$87$	0	0
$88$	1502.68	1.82030
$89$	−1012.63	−1.20606	−0.603029	−	0.797720i	$-0.706040\pi$
$89$	−1012.63	−1.20606	−0.603029	+	0.797720i	$0.706040\pi$
$90$	0	0
$91$	−1269.33	−1.46222
$92$	−11.2487	−0.0127474
$93$	0	0
$94$	−1146.77	−1.25831
$95$	192.867	0.208292
$96$	0	0
$97$	−397.912	−0.416513	−0.208257	−	0.978074i	$-0.566779\pi$
$97$	−397.912	−0.416513	−0.208257	+	0.978074i	$0.566779\pi$
$98$	−152.500	−0.157192
$99$	0	0
$100$	70.2212	0.0702212
$101$	961.752	0.947504	0.473752	−	0.880658i	$-0.342899\pi$
$101$	961.752	0.947504	0.473752	+	0.880658i	$0.342899\pi$
$102$	0	0
$103$	−1085.47	−1.03839	−0.519197	−	0.854654i	$-0.673769\pi$
$103$	−1085.47	−1.03839	−0.519197	+	0.854654i	$0.673769\pi$
$104$	1800.76	1.69788
$105$	0	0
$106$	926.279	0.848757
$107$	−39.2036	−0.0354201	−0.0177101	−	0.999843i	$-0.505638\pi$
$107$	−39.2036	−0.0354201	−0.0177101	+	0.999843i	$0.505638\pi$
$108$	0	0
$109$	−210.156	−0.184672	−0.0923361	−	0.995728i	$-0.529433\pi$
$109$	−210.156	−0.184672	−0.0923361	+	0.995728i	$0.529433\pi$
$110$	−1309.63	−1.13516
$111$	0	0
$112$	−903.958	−0.762643
$113$	889.542	0.740540	0.370270	−	0.928924i	$-0.379265\pi$
$113$	889.542	0.740540	0.370270	+	0.928924i	$0.379265\pi$
$114$	0	0
$115$	78.5245	0.0636734
$116$	182.994	0.146471
$117$	0	0
$118$	−154.516	−0.120546
$119$	877.175	0.675719
$120$	0	0
$121$	2608.84	1.96006
$122$	−1826.24	−1.35524
$123$	0	0
$124$	143.640	0.104026
$125$	−1486.05	−1.06333
$126$	0	0
$127$	−2040.11	−1.42543	−0.712717	−	0.701451i	$-0.752536\pi$
$127$	−2040.11	−1.42543	−0.712717	+	0.701451i	$0.752536\pi$
$128$	1063.84	0.734618
$129$	0	0
$130$	−1569.41	−1.05882
$131$	2233.61	1.48971	0.744853	−	0.667228i	$-0.232519\pi$
$131$	2233.61	1.48971	0.744853	+	0.667228i	$0.232519\pi$
$132$	0	0
$133$	408.525	0.266343
$134$	−764.792	−0.493045
$135$	0	0
$136$	−1244.42	−0.784620
$137$	2125.70	1.32563	0.662815	−	0.748784i	$-0.269362\pi$
$137$	2125.70	1.32563	0.662815	+	0.748784i	$0.269362\pi$
$138$	0	0
$139$	2710.99	1.65427	0.827133	−	0.562006i	$-0.189970\pi$
$139$	2710.99	1.65427	0.827133	+	0.562006i	$0.189970\pi$
$140$	−153.432	−0.0926244
$141$	0	0
$142$	−585.318	−0.345907
$143$	4721.36	2.76098
$144$	0	0
$145$	−1277.44	−0.731622
$146$	321.603	0.182301
$147$	0	0
$148$	255.799	0.142071
$149$	−1041.19	−0.572470	−0.286235	−	0.958160i	$-0.592404\pi$
$149$	−1041.19	−0.572470	−0.286235	+	0.958160i	$0.592404\pi$
$150$	0	0
$151$	1448.13	0.780444	0.390222	−	0.920721i	$-0.372398\pi$
$151$	1448.13	0.780444	0.390222	+	0.920721i	$0.372398\pi$
$152$	−579.562	−0.309267
$153$	0	0
$154$	−2774.02	−1.45154
$155$	−1002.71	−0.519612
$156$	0	0
$157$	−2438.90	−1.23978	−0.619891	−	0.784688i	$-0.712823\pi$
$157$	−2438.90	−1.23978	−0.619891	+	0.784688i	$0.712823\pi$
$158$	−1153.74	−0.580928
$159$	0	0
$160$	408.164	0.201676
$161$	166.328	0.0814193
$162$	0	0
$163$	3332.07	1.60115	0.800575	−	0.599232i	$-0.204527\pi$
$163$	3332.07	1.60115	0.800575	+	0.599232i	$0.204527\pi$
$164$	−533.507	−0.254024
$165$	0	0
$166$	1734.68	0.811068
$167$	2643.67	1.22499	0.612494	−	0.790475i	$-0.290166\pi$
$167$	2643.67	1.22499	0.612494	+	0.790475i	$0.290166\pi$
$168$	0	0
$169$	3460.91	1.57529
$170$	1084.54	0.489299
$171$	0	0
$172$	−103.810	−0.0460200
$173$	1207.93	0.530849	0.265424	−	0.964132i	$-0.414488\pi$
$173$	1207.93	0.530849	0.265424	+	0.964132i	$0.414488\pi$
$174$	0	0
$175$	−1038.32	−0.448512
$176$	3362.33	1.44003
$177$	0	0
$178$	−2652.01	−1.11672
$179$	1379.61	0.576074	0.288037	−	0.957619i	$-0.406997\pi$
$179$	1379.61	0.576074	0.288037	+	0.957619i	$0.406997\pi$
$180$	0	0
$181$	−2223.79	−0.913221	−0.456611	−	0.889667i	$-0.650937\pi$
$181$	−2223.79	−0.913221	−0.456611	+	0.889667i	$0.650937\pi$
$182$	−3324.28	−1.35391
$183$	0	0
$184$	−235.965	−0.0945411
$185$	−1785.67	−0.709648
$186$	0	0
$187$	−3262.71	−1.27590
$188$	499.735	0.193867
$189$	0	0
$190$	505.102	0.192863
$191$	1419.51	0.537762	0.268881	−	0.963173i	$-0.413346\pi$
$191$	1419.51	0.537762	0.268881	+	0.963173i	$0.413346\pi$
$192$	0	0
$193$	2797.83	1.04348	0.521741	−	0.853104i	$-0.325283\pi$
$193$	2797.83	1.04348	0.521741	+	0.853104i	$0.325283\pi$
$194$	−1042.10	−0.385661
$195$	0	0
$196$	66.4556	0.0242185
$197$	−4134.50	−1.49528	−0.747641	−	0.664103i	$-0.768814\pi$
$197$	−4134.50	−1.49528	−0.747641	+	0.664103i	$0.768814\pi$
$198$	0	0
$199$	−5049.42	−1.79871	−0.899357	−	0.437214i	$-0.855965\pi$
$199$	−5049.42	−1.79871	−0.899357	+	0.437214i	$0.855965\pi$
$200$	1473.03	0.520795
$201$	0	0
$202$	2518.75	0.877320
$203$	−2705.83	−0.935527
$204$	0	0
$205$	3724.28	1.26885
$206$	−2842.76	−0.961479
$207$	0	0
$208$	4029.29	1.34318
$209$	−1519.54	−0.502911
$210$	0	0
$211$	4910.79	1.60224	0.801121	−	0.598503i	$-0.204237\pi$
$211$	4910.79	1.60224	0.801121	+	0.598503i	$0.204237\pi$
$212$	−403.649	−0.130768
$213$	0	0
$214$	−102.671	−0.0327965
$215$	724.670	0.229870
$216$	0	0
$217$	−2123.92	−0.664429
$218$	−550.381	−0.170993
$219$	0	0
$220$	570.702	0.174894
$221$	−3909.91	−1.19009
$222$	0	0
$223$	−47.4526	−0.0142496	−0.00712480	−	0.999975i	$-0.502268\pi$
$223$	−47.4526	−0.0142496	−0.00712480	+	0.999975i	$0.502268\pi$
$224$	864.562	0.257884
$225$	0	0
$226$	2329.64	0.685687
$227$	−3097.71	−0.905737	−0.452869	−	0.891577i	$-0.649599\pi$
$227$	−3097.71	−0.905737	−0.452869	+	0.891577i	$0.649599\pi$
$228$	0	0
$229$	2273.82	0.656149	0.328074	−	0.944652i	$-0.393600\pi$
$229$	2273.82	0.656149	0.328074	+	0.944652i	$0.393600\pi$
$230$	205.649	0.0589570
$231$	0	0
$232$	3838.68	1.08630
$233$	3433.68	0.965442	0.482721	−	0.875774i	$-0.339649\pi$
$233$	3433.68	0.965442	0.482721	+	0.875774i	$0.339649\pi$
$234$	0	0
$235$	−3488.52	−0.968367
$236$	67.3343	0.0185724
$237$	0	0
$238$	2297.25	0.625667
$239$	−5489.29	−1.48566	−0.742830	−	0.669480i	$-0.766517\pi$
$239$	−5489.29	−1.48566	−0.742830	+	0.669480i	$0.766517\pi$
$240$	0	0
$241$	71.4197	0.0190894	0.00954470	−	0.999954i	$-0.496962\pi$
$241$	71.4197	0.0190894	0.00954470	+	0.999954i	$0.496962\pi$
$242$	6832.35	1.81488
$243$	0	0
$244$	795.829	0.208802
$245$	−463.909	−0.120972
$246$	0	0
$247$	−1820.96	−0.469088
$248$	3013.14	0.771511
$249$	0	0
$250$	−3891.85	−0.984568
$251$	−7078.52	−1.78005	−0.890024	−	0.455913i	$-0.849313\pi$
$251$	−7078.52	−1.78005	−0.890024	+	0.455913i	$0.849313\pi$
$252$	0	0
$253$	−618.670	−0.153737
$254$	−5342.88	−1.31985
$255$	0	0
$256$	−1715.60	−0.418848
$257$	5131.54	1.24551	0.622756	−	0.782416i	$-0.286013\pi$
$257$	5131.54	1.24551	0.622756	+	0.782416i	$0.286013\pi$
$258$	0	0
$259$	−3782.35	−0.907428
$260$	683.909	0.163132
$261$	0	0
$262$	5849.65	1.37936
$263$	2899.64	0.679846	0.339923	−	0.940453i	$-0.389599\pi$
$263$	2899.64	0.679846	0.339923	+	0.940453i	$0.389599\pi$
$264$	0	0
$265$	2817.77	0.653186
$266$	1069.89	0.246614
$267$	0	0
$268$	333.277	0.0759632
$269$	−2624.44	−0.594852	−0.297426	−	0.954745i	$-0.596128\pi$
$269$	−2624.44	−0.594852	−0.297426	+	0.954745i	$0.596128\pi$
$270$	0	0
$271$	8503.68	1.90613	0.953066	−	0.302762i	$-0.0979088\pi$
$271$	8503.68	1.90613	0.953066	+	0.302762i	$0.0979088\pi$
$272$	−2784.46	−0.620707
$273$	0	0
$274$	5567.05	1.22744
$275$	3862.10	0.846884
$276$	0	0
$277$	−3533.33	−0.766415	−0.383208	−	0.923662i	$-0.625181\pi$
$277$	−3533.33	−0.766415	−0.383208	+	0.923662i	$0.625181\pi$
$278$	7099.86	1.53173
$279$	0	0
$280$	−3218.56	−0.686948
$281$	2767.76	0.587584	0.293792	−	0.955869i	$-0.405083\pi$
$281$	2767.76	0.587584	0.293792	+	0.955869i	$0.405083\pi$
$282$	0	0
$283$	−3853.33	−0.809388	−0.404694	−	0.914452i	$-0.632622\pi$
$283$	−3853.33	−0.809388	−0.404694	+	0.914452i	$0.632622\pi$
$284$	255.067	0.0532938
$285$	0	0
$286$	12364.9	2.55647
$287$	7888.66	1.62248
$288$	0	0
$289$	−2211.04	−0.450039
$290$	−3345.50	−0.677430
$291$	0	0
$292$	−140.146	−0.0280871
$293$	−6720.24	−1.33993	−0.669967	−	0.742391i	$-0.733692\pi$
$293$	−6720.24	−1.33993	−0.669967	+	0.742391i	$0.733692\pi$
$294$	0	0
$295$	−470.043	−0.0927694
$296$	5365.90	1.05367
$297$	0	0
$298$	−2726.80	−0.530066
$299$	−741.391	−0.143397
$300$	0	0
$301$	1534.98	0.293935
$302$	3792.53	0.722635
$303$	0	0
$304$	−1296.80	−0.244659
$305$	−5555.47	−1.04297
$306$	0	0
$307$	−3154.37	−0.586416	−0.293208	−	0.956049i	$-0.594723\pi$
$307$	−3154.37	−0.586416	−0.293208	+	0.956049i	$0.594723\pi$
$308$	1208.85	0.223638
$309$	0	0
$310$	−2626.03	−0.481124
$311$	−4204.57	−0.766622	−0.383311	−	0.923619i	$-0.625216\pi$
$311$	−4204.57	−0.766622	−0.383311	+	0.923619i	$0.625216\pi$
$312$	0	0
$313$	−6497.54	−1.17336	−0.586682	−	0.809818i	$-0.699566\pi$
$313$	−6497.54	−1.17336	−0.586682	+	0.809818i	$0.699566\pi$
$314$	−6387.29	−1.14795
$315$	0	0
$316$	502.771	0.0895034
$317$	−5621.97	−0.996092	−0.498046	−	0.867151i	$-0.665949\pi$
$317$	−5621.97	−0.996092	−0.498046	+	0.867151i	$0.665949\pi$
$318$	0	0
$319$	10064.5	1.76647
$320$	4483.05	0.783157
$321$	0	0
$322$	435.601	0.0753884
$323$	1258.38	0.216774
$324$	0	0
$325$	4628.19	0.789926
$326$	8726.41	1.48255
$327$	0	0
$328$	−11191.4	−1.88397
$329$	−7389.29	−1.23825
$330$	0	0
$331$	−7641.62	−1.26895	−0.634473	−	0.772945i	$-0.718783\pi$
$331$	−7641.62	−1.26895	−0.634473	+	0.772945i	$0.718783\pi$
$332$	−755.930	−0.124961
$333$	0	0
$334$	6923.55	1.13425
$335$	−2326.52	−0.379437
$336$	0	0
$337$	7138.77	1.15393	0.576964	−	0.816770i	$-0.304237\pi$
$337$	7138.77	1.15393	0.576964	+	0.816770i	$0.304237\pi$
$338$	9063.84	1.45860
$339$	0	0
$340$	−472.617	−0.0753861
$341$	7900.07	1.25458
$342$	0	0
$343$	−6770.81	−1.06586
$344$	−2177.62	−0.341307
$345$	0	0
$346$	3163.46	0.491528
$347$	−3416.69	−0.528580	−0.264290	−	0.964443i	$-0.585138\pi$
$347$	−3416.69	−0.528580	−0.264290	+	0.964443i	$0.585138\pi$
$348$	0	0
$349$	8930.15	1.36968	0.684842	−	0.728691i	$-0.259871\pi$
$349$	8930.15	1.36968	0.684842	+	0.728691i	$0.259871\pi$
$350$	−2719.27	−0.415289
$351$	0	0
$352$	−3215.80	−0.486939
$353$	−9550.53	−1.44001	−0.720005	−	0.693969i	$-0.755861\pi$
$353$	−9550.53	−1.44001	−0.720005	+	0.693969i	$0.755861\pi$
$354$	0	0
$355$	−1780.56	−0.266203
$356$	1155.68	0.172053
$357$	0	0
$358$	3613.10	0.533403
$359$	7990.13	1.17466	0.587330	−	0.809347i	$-0.300179\pi$
$359$	7990.13	1.17466	0.587330	+	0.809347i	$0.300179\pi$
$360$	0	0
$361$	−6272.94	−0.914556
$362$	−5823.93	−0.845577
$363$	0	0
$364$	1448.64	0.208597
$365$	978.325	0.140296
$366$	0	0
$367$	−9255.46	−1.31643	−0.658216	−	0.752829i	$-0.728689\pi$
$367$	−9255.46	−1.31643	−0.658216	+	0.752829i	$0.728689\pi$
$368$	−527.983	−0.0747908
$369$	0	0
$370$	−4676.52	−0.657083
$371$	5968.52	0.835230
$372$	0	0
$373$	−5532.87	−0.768047	−0.384023	−	0.923323i	$-0.625462\pi$
$373$	−5532.87	−0.768047	−0.384023	+	0.923323i	$0.625462\pi$
$374$	−8544.78	−1.18139
$375$	0	0
$376$	10483.0	1.43781
$377$	12060.9	1.64767
$378$	0	0
$379$	12355.4	1.67456	0.837278	−	0.546778i	$-0.184146\pi$
$379$	12355.4	1.67456	0.837278	+	0.546778i	$0.184146\pi$
$380$	−220.111	−0.0297143
$381$	0	0
$382$	3717.59	0.497929
$383$	12814.3	1.70961	0.854807	−	0.518946i	$-0.173675\pi$
$383$	12814.3	1.70961	0.854807	+	0.518946i	$0.173675\pi$
$384$	0	0
$385$	−8438.64	−1.11707
$386$	7327.28	0.966188
$387$	0	0
$388$	454.120	0.0594187
$389$	1715.48	0.223595	0.111797	−	0.993731i	$-0.464339\pi$
$389$	1715.48	0.223595	0.111797	+	0.993731i	$0.464339\pi$
$390$	0	0
$391$	512.340	0.0662664
$392$	1394.04	0.179617
$393$	0	0
$394$	−10827.9	−1.38452
$395$	−3509.71	−0.447071
$396$	0	0
$397$	1097.81	0.138785	0.0693924	−	0.997589i	$-0.477894\pi$
$397$	1097.81	0.138785	0.0693924	+	0.997589i	$0.477894\pi$
$398$	−13224.0	−1.66548
$399$	0	0
$400$	3295.98	0.411998
$401$	3949.81	0.491880	0.245940	−	0.969285i	$-0.420903\pi$
$401$	3949.81	0.491880	0.245940	+	0.969285i	$0.420903\pi$
$402$	0	0
$403$	9467.15	1.17020
$404$	−1097.61	−0.135168
$405$	0	0
$406$	−7086.35	−0.866230
$407$	14068.7	1.71342
$408$	0	0
$409$	−1709.43	−0.206665	−0.103332	−	0.994647i	$-0.532951\pi$
$409$	−1709.43	−0.206665	−0.103332	+	0.994647i	$0.532951\pi$
$410$	9753.58	1.17487
$411$	0	0
$412$	1238.80	0.148135
$413$	−995.633	−0.118624
$414$	0	0
$415$	5276.95	0.624181
$416$	−3853.69	−0.454190
$417$	0	0
$418$	−3979.54	−0.465660
$419$	280.723	0.0327308	0.0163654	−	0.999866i	$-0.494791\pi$
$419$	280.723	0.0327308	0.0163654	+	0.999866i	$0.494791\pi$
$420$	0	0
$421$	−11642.2	−1.34776	−0.673878	−	0.738842i	$-0.735373\pi$
$421$	−11642.2	−1.34776	−0.673878	+	0.738842i	$0.735373\pi$
$422$	12861.0	1.48356
$423$	0	0
$424$	−8467.36	−0.969838
$425$	−3198.33	−0.365039
$426$	0	0
$427$	−11767.5	−1.33365
$428$	44.7414	0.00505294
$429$	0	0
$430$	1897.85	0.212843
$431$	1887.48	0.210943	0.105472	−	0.994422i	$-0.466365\pi$
$431$	1887.48	0.210943	0.105472	+	0.994422i	$0.466365\pi$
$432$	0	0
$433$	−2668.89	−0.296210	−0.148105	−	0.988972i	$-0.547317\pi$
$433$	−2668.89	−0.296210	−0.148105	+	0.988972i	$0.547317\pi$
$434$	−5562.38	−0.615213
$435$	0	0
$436$	239.842	0.0263448
$437$	238.611	0.0261197
$438$	0	0
$439$	11389.6	1.23826	0.619131	−	0.785287i	$-0.287485\pi$
$439$	11389.6	1.23826	0.619131	+	0.785287i	$0.287485\pi$
$440$	11971.6	1.29710
$441$	0	0
$442$	−10239.8	−1.10193
$443$	−14630.8	−1.56914	−0.784569	−	0.620041i	$-0.787116\pi$
$443$	−14630.8	−1.56914	−0.784569	+	0.620041i	$0.787116\pi$
$444$	0	0
$445$	−8067.50	−0.859407
$446$	−124.274	−0.0131941
$447$	0	0
$448$	9495.88	1.00142
$449$	−10838.9	−1.13924	−0.569620	−	0.821908i	$-0.692910\pi$
$449$	−10838.9	−1.13924	−0.569620	+	0.821908i	$0.692910\pi$
$450$	0	0
$451$	−29342.4	−3.06359
$452$	−1015.20	−0.105644
$453$	0	0
$454$	−8112.66	−0.838648
$455$	−10112.6	−1.04194
$456$	0	0
$457$	−7768.54	−0.795179	−0.397590	−	0.917563i	$-0.630153\pi$
$457$	−7768.54	−0.795179	−0.397590	+	0.917563i	$0.630153\pi$
$458$	5954.95	0.607547
$459$	0	0
$460$	−89.6168	−0.00908349
$461$	−7262.50	−0.733727	−0.366864	−	0.930275i	$-0.619568\pi$
$461$	−7262.50	−0.733727	−0.366864	+	0.930275i	$0.619568\pi$
$462$	0	0
$463$	8935.85	0.896942	0.448471	−	0.893797i	$-0.351969\pi$
$463$	8935.85	0.896942	0.448471	+	0.893797i	$0.351969\pi$
$464$	8589.23	0.859364
$465$	0	0
$466$	8992.53	0.893929
$467$	−17921.0	−1.77577	−0.887885	−	0.460065i	$-0.847826\pi$
$467$	−17921.0	−1.77577	−0.887885	+	0.460065i	$0.847826\pi$
$468$	0	0
$469$	−4927.97	−0.485187
$470$	−9136.16	−0.896638
$471$	0	0
$472$	1412.47	0.137742
$473$	−5709.45	−0.555012
$474$	0	0
$475$	−1489.55	−0.143885
$476$	−1001.08	−0.0963963
$477$	0	0
$478$	−14376.0	−1.37561
$479$	−1733.15	−0.165323	−0.0826616	−	0.996578i	$-0.526342\pi$
$479$	−1733.15	−0.165323	−0.0826616	+	0.996578i	$0.526342\pi$
$480$	0	0
$481$	16859.4	1.59818
$482$	187.042	0.0176754
$483$	0	0
$484$	−2977.37	−0.279618
$485$	−3170.10	−0.296797
$486$	0	0
$487$	−2597.17	−0.241661	−0.120831	−	0.992673i	$-0.538556\pi$
$487$	−2597.17	−0.241661	−0.120831	+	0.992673i	$0.538556\pi$
$488$	16694.1	1.54858
$489$	0	0
$490$	−1214.94	−0.112011
$491$	−7523.32	−0.691492	−0.345746	−	0.938328i	$-0.612374\pi$
$491$	−7523.32	−0.691492	−0.345746	+	0.938328i	$0.612374\pi$
$492$	0	0
$493$	−8334.75	−0.761416
$494$	−4768.94	−0.434341
$495$	0	0
$496$	6742.05	0.610337
$497$	−3771.52	−0.340394
$498$	0	0
$499$	9392.79	0.842643	0.421322	−	0.906911i	$-0.361566\pi$
$499$	9392.79	0.842643	0.421322	+	0.906911i	$0.361566\pi$
$500$	1695.97	0.151692
$501$	0	0
$502$	−18538.1	−1.64820
$503$	−2687.97	−0.238272	−0.119136	−	0.992878i	$-0.538012\pi$
$503$	−2687.97	−0.238272	−0.119136	+	0.992878i	$0.538012\pi$
$504$	0	0
$505$	7662.12	0.675168
$506$	−1620.25	−0.142349
$507$	0	0
$508$	2328.29	0.203349
$509$	−10267.4	−0.894098	−0.447049	−	0.894510i	$-0.647525\pi$
$509$	−10267.4	−0.894098	−0.447049	+	0.894510i	$0.647525\pi$
$510$	0	0
$511$	2072.26	0.179396
$512$	−13003.8	−1.12244
$513$	0	0
$514$	13439.1	1.15325
$515$	−8647.77	−0.739935
$516$	0	0
$517$	27485.0	2.33808
$518$	−9905.67	−0.840213
$519$	0	0
$520$	14346.4	1.20987
$521$	14725.4	1.23826	0.619128	−	0.785290i	$-0.287486\pi$
$521$	14725.4	1.23826	0.619128	+	0.785290i	$0.287486\pi$
$522$	0	0
$523$	5672.69	0.474282	0.237141	−	0.971475i	$-0.423790\pi$
$523$	5672.69	0.474282	0.237141	+	0.971475i	$0.423790\pi$
$524$	−2549.13	−0.212518
$525$	0	0
$526$	7593.93	0.629488
$527$	−6542.30	−0.540772
$528$	0	0
$529$	−12069.9	−0.992015
$530$	7379.52	0.604803
$531$	0	0
$532$	−466.233	−0.0379958
$533$	−35162.8	−2.85755
$534$	0	0
$535$	−312.328	−0.0252395
$536$	6991.17	0.563381
$537$	0	0
$538$	−6873.20	−0.550790
$539$	3655.00	0.292081
$540$	0	0
$541$	−13483.4	−1.07153	−0.535766	−	0.844367i	$-0.679977\pi$
$541$	−13483.4	−1.07153	−0.535766	+	0.844367i	$0.679977\pi$
$542$	22270.5	1.76494
$543$	0	0
$544$	2663.11	0.209889
$545$	−1674.28	−0.131593
$546$	0	0
$547$	3503.56	0.273860	0.136930	−	0.990581i	$-0.456277\pi$
$547$	3503.56	0.273860	0.136930	+	0.990581i	$0.456277\pi$
$548$	−2425.98	−0.189111
$549$	0	0
$550$	10114.5	0.784154
$551$	−3881.72	−0.300122
$552$	0	0
$553$	−7434.18	−0.571670
$554$	−9253.50	−0.709645
$555$	0	0
$556$	−3093.94	−0.235993
$557$	−785.734	−0.0597713	−0.0298856	−	0.999553i	$-0.509514\pi$
$557$	−785.734	−0.0597713	−0.0298856	+	0.999553i	$0.509514\pi$
$558$	0	0
$559$	−6841.99	−0.517684
$560$	−7201.68	−0.543440
$561$	0	0
$562$	7248.55	0.544060
$563$	711.755	0.0532805	0.0266402	−	0.999645i	$-0.491519\pi$
$563$	711.755	0.0532805	0.0266402	+	0.999645i	$0.491519\pi$
$564$	0	0
$565$	7086.83	0.527691
$566$	−10091.6	−0.749435
$567$	0	0
$568$	5350.54	0.395253
$569$	12752.6	0.939574	0.469787	−	0.882780i	$-0.344331\pi$
$569$	12752.6	0.939574	0.469787	+	0.882780i	$0.344331\pi$
$570$	0	0
$571$	568.844	0.0416907	0.0208454	−	0.999783i	$-0.493364\pi$
$571$	568.844	0.0416907	0.0208454	+	0.999783i	$0.493364\pi$
$572$	−5388.30	−0.393874
$573$	0	0
$574$	20659.8	1.50230
$575$	−606.461	−0.0439846
$576$	0	0
$577$	−4814.50	−0.347366	−0.173683	−	0.984802i	$-0.555567\pi$
$577$	−4814.50	−0.347366	−0.173683	+	0.984802i	$0.555567\pi$
$578$	−5790.54	−0.416704
$579$	0	0
$580$	1457.89	0.104371
$581$	11177.5	0.798142
$582$	0	0
$583$	−22200.3	−1.57709
$584$	−2939.85	−0.208308
$585$	0	0
$586$	−17599.8	−1.24068
$587$	3790.32	0.266513	0.133256	−	0.991082i	$-0.457457\pi$
$587$	3790.32	0.266513	0.133256	+	0.991082i	$0.457457\pi$
$588$	0	0
$589$	−3046.93	−0.213152
$590$	−1231.01	−0.0858978
$591$	0	0
$592$	12006.5	0.833553
$593$	−15605.2	−1.08065	−0.540326	−	0.841456i	$-0.681699\pi$
$593$	−15605.2	−1.08065	−0.540326	+	0.841456i	$0.681699\pi$
$594$	0	0
$595$	6988.31	0.481500
$596$	1188.27	0.0816670
$597$	0	0
$598$	−1941.64	−0.132775
$599$	26016.3	1.77462	0.887310	−	0.461174i	$-0.152572\pi$
$599$	26016.3	1.77462	0.887310	+	0.461174i	$0.152572\pi$
$600$	0	0
$601$	−15990.4	−1.08530	−0.542648	−	0.839960i	$-0.682578\pi$
$601$	−15990.4	−1.08530	−0.542648	+	0.839960i	$0.682578\pi$
$602$	4019.98	0.272163
$603$	0	0
$604$	−1652.69	−0.111336
$605$	20784.2	1.39669
$606$	0	0
$607$	8461.94	0.565831	0.282915	−	0.959145i	$-0.408698\pi$
$607$	8461.94	0.565831	0.282915	+	0.959145i	$0.408698\pi$
$608$	1240.28	0.0827304
$609$	0	0
$610$	−14549.3	−0.965714
$611$	32937.0	2.18083
$612$	0	0
$613$	−3306.42	−0.217855	−0.108927	−	0.994050i	$-0.534742\pi$
$613$	−3306.42	−0.217855	−0.108927	+	0.994050i	$0.534742\pi$
$614$	−8261.05	−0.542979
$615$	0	0
$616$	25358.0	1.65861
$617$	2038.03	0.132979	0.0664894	−	0.997787i	$-0.478820\pi$
$617$	2038.03	0.132979	0.0664894	+	0.997787i	$0.478820\pi$
$618$	0	0
$619$	−1376.65	−0.0893896	−0.0446948	−	0.999001i	$-0.514232\pi$
$619$	−1376.65	−0.0893896	−0.0446948	+	0.999001i	$0.514232\pi$
$620$	1144.36	0.0741266
$621$	0	0
$622$	−11011.4	−0.709837
$623$	−17088.3	−1.09892
$624$	0	0
$625$	−4147.92	−0.265467
$626$	−17016.5	−1.08645
$627$	0	0
$628$	2783.42	0.176864
$629$	−11650.8	−0.738547
$630$	0	0
$631$	−12856.2	−0.811091	−0.405546	−	0.914075i	$-0.632918\pi$
$631$	−12856.2	−0.811091	−0.405546	+	0.914075i	$0.632918\pi$
$632$	10546.6	0.663802
$633$	0	0
$634$	−14723.5	−0.922309
$635$	−16253.2	−1.01573
$636$	0	0
$637$	4380.01	0.272437
$638$	26358.2	1.63563
$639$	0	0
$640$	8475.44	0.523471
$641$	−6936.45	−0.427416	−0.213708	−	0.976898i	$-0.568554\pi$
$641$	−6936.45	−0.427416	−0.213708	+	0.976898i	$0.568554\pi$
$642$	0	0
$643$	−30055.9	−1.84337	−0.921685	−	0.387938i	$-0.873187\pi$
$643$	−30055.9	−1.84337	−0.921685	+	0.387938i	$0.873187\pi$
$644$	−189.824	−0.0116151
$645$	0	0
$646$	3295.59	0.200717
$647$	22193.5	1.34856	0.674280	−	0.738476i	$-0.264454\pi$
$647$	22193.5	1.34856	0.674280	+	0.738476i	$0.264454\pi$
$648$	0	0
$649$	3703.32	0.223988
$650$	12120.9	0.731415
$651$	0	0
$652$	−3802.75	−0.228416
$653$	4334.02	0.259729	0.129865	−	0.991532i	$-0.458546\pi$
$653$	4334.02	0.259729	0.129865	+	0.991532i	$0.458546\pi$
$654$	0	0
$655$	17794.8	1.06153
$656$	−25041.3	−1.49039
$657$	0	0
$658$	−19352.0	−1.14653
$659$	−10735.4	−0.634584	−0.317292	−	0.948328i	$-0.602774\pi$
$659$	−10735.4	−0.634584	−0.317292	+	0.948328i	$0.602774\pi$
$660$	0	0
$661$	−31199.2	−1.83587	−0.917934	−	0.396734i	$-0.870144\pi$
$661$	−31199.2	−1.83587	−0.917934	+	0.396734i	$0.870144\pi$
$662$	−20012.8	−1.17495
$663$	0	0
$664$	−15857.2	−0.926773
$665$	3254.65	0.189789
$666$	0	0
$667$	−1580.42	−0.0917453
$668$	−3017.11	−0.174754
$669$	0	0
$670$	−6092.97	−0.351332
$671$	43769.8	2.51821
$672$	0	0
$673$	4517.81	0.258765	0.129383	−	0.991595i	$-0.458700\pi$
$673$	4517.81	0.258765	0.129383	+	0.991595i	$0.458700\pi$
$674$	18695.9	1.06845
$675$	0	0
$676$	−3949.79	−0.224726
$677$	−4980.54	−0.282744	−0.141372	−	0.989957i	$-0.545151\pi$
$677$	−4980.54	−0.282744	−0.141372	+	0.989957i	$0.545151\pi$
$678$	0	0
$679$	−6714.81	−0.379515
$680$	−9914.10	−0.559101
$681$	0	0
$682$	20689.6	1.16165
$683$	11878.0	0.665446	0.332723	−	0.943025i	$-0.392033\pi$
$683$	11878.0	0.665446	0.332723	+	0.943025i	$0.392033\pi$
$684$	0	0
$685$	16935.1	0.944611
$686$	−17732.2	−0.986908
$687$	0	0
$688$	−4872.54	−0.270006
$689$	−26604.1	−1.47102
$690$	0	0
$691$	3069.47	0.168984	0.0844921	−	0.996424i	$-0.473073\pi$
$691$	3069.47	0.168984	0.0844921	+	0.996424i	$0.473073\pi$
$692$	−1378.56	−0.0757295
$693$	0	0
$694$	−8948.03	−0.489427
$695$	21598.0	1.17879
$696$	0	0
$697$	24299.4	1.32052
$698$	23387.3	1.26823
$699$	0	0
$700$	1184.99	0.0639835
$701$	14816.8	0.798321	0.399161	−	0.916881i	$-0.369302\pi$
$701$	14816.8	0.798321	0.399161	+	0.916881i	$0.369302\pi$
$702$	0	0
$703$	−5426.08	−0.291107
$704$	−35320.6	−1.89090
$705$	0	0
$706$	−25012.1	−1.33335
$707$	16229.7	0.863338
$708$	0	0
$709$	29464.2	1.56072	0.780361	−	0.625329i	$-0.215035\pi$
$709$	29464.2	1.56072	0.780361	+	0.625329i	$0.215035\pi$
$710$	−4663.13	−0.246485
$711$	0	0
$712$	24242.7	1.27603
$713$	−1240.54	−0.0651592
$714$	0	0
$715$	37614.3	1.96741
$716$	−1574.50	−0.0821812
$717$	0	0
$718$	20925.5	1.08765
$719$	−16747.1	−0.868654	−0.434327	−	0.900755i	$-0.643014\pi$
$719$	−16747.1	−0.868654	−0.434327	+	0.900755i	$0.643014\pi$
$720$	0	0
$721$	−18317.5	−0.946156
$722$	−16428.3	−0.846813
$723$	0	0
$724$	2537.92	0.130278
$725$	9865.90	0.505394
$726$	0	0
$727$	13485.0	0.687936	0.343968	−	0.938981i	$-0.388229\pi$
$727$	13485.0	0.687936	0.343968	+	0.938981i	$0.388229\pi$
$728$	30388.1	1.54706
$729$	0	0
$730$	2562.15	0.129904
$731$	4728.18	0.239231
$732$	0	0
$733$	−25581.8	−1.28906	−0.644532	−	0.764577i	$-0.722948\pi$
$733$	−25581.8	−1.28906	−0.644532	+	0.764577i	$0.722948\pi$
$734$	−24239.3	−1.21892
$735$	0	0
$736$	504.973	0.0252902
$737$	18329.9	0.916135
$738$	0	0
$739$	−8290.39	−0.412675	−0.206338	−	0.978481i	$-0.566155\pi$
$739$	−8290.39	−0.412675	−0.206338	+	0.978481i	$0.566155\pi$
$740$	2037.91	0.101237
$741$	0	0
$742$	15631.1	0.773363
$743$	29226.2	1.44308	0.721539	−	0.692374i	$-0.243435\pi$
$743$	29226.2	1.44308	0.721539	+	0.692374i	$0.243435\pi$
$744$	0	0
$745$	−8295.02	−0.407928
$746$	−14490.2	−0.711156
$747$	0	0
$748$	3723.60	0.182016
$749$	−661.565	−0.0322738
$750$	0	0
$751$	4511.16	0.219194	0.109597	−	0.993976i	$-0.465044\pi$
$751$	4511.16	0.219194	0.109597	+	0.993976i	$0.465044\pi$
$752$	23456.2	1.13744
$753$	0	0
$754$	31586.6	1.52562
$755$	11537.0	0.556125
$756$	0	0
$757$	−14064.1	−0.675256	−0.337628	−	0.941280i	$-0.609625\pi$
$757$	−14064.1	−0.675256	−0.337628	+	0.941280i	$0.609625\pi$
$758$	32357.9	1.55052
$759$	0	0
$760$	−4617.27	−0.220376
$761$	31754.7	1.51262	0.756312	−	0.654211i	$-0.226999\pi$
$761$	31754.7	1.51262	0.756312	+	0.654211i	$0.226999\pi$
$762$	0	0
$763$	−3546.40	−0.168268
$764$	−1620.03	−0.0767157
$765$	0	0
$766$	33559.7	1.58298
$767$	4437.92	0.208923
$768$	0	0
$769$	23548.0	1.10424	0.552121	−	0.833764i	$-0.313819\pi$
$769$	23548.0	1.10424	0.552121	+	0.833764i	$0.313819\pi$
$770$	−22100.1	−1.03433
$771$	0	0
$772$	−3193.04	−0.148860
$773$	14079.0	0.655092	0.327546	−	0.944835i	$-0.393778\pi$
$773$	14079.0	0.655092	0.327546	+	0.944835i	$0.393778\pi$
$774$	0	0
$775$	7744.17	0.358940
$776$	9526.09	0.440679
$777$	0	0
$778$	4492.71	0.207033
$779$	11316.9	0.520501
$780$	0	0
$781$	14028.4	0.642736
$782$	1341.78	0.0613579
$783$	0	0
$784$	3119.24	0.142093
$785$	−19430.3	−0.883438
$786$	0	0
$787$	−11840.6	−0.536305	−0.268152	−	0.963377i	$-0.586413\pi$
$787$	−11840.6	−0.536305	−0.268152	+	0.963377i	$0.586413\pi$
$788$	4718.53	0.213313
$789$	0	0
$790$	−9191.66	−0.413955
$791$	15011.1	0.674759
$792$	0	0
$793$	52452.1	2.34884
$794$	2875.08	0.128505
$795$	0	0
$796$	5762.70	0.256600
$797$	−4057.07	−0.180312	−0.0901561	−	0.995928i	$-0.528737\pi$
$797$	−4057.07	−0.180312	−0.0901561	+	0.995928i	$0.528737\pi$
$798$	0	0
$799$	−22761.2	−1.00780
$800$	−3152.34	−0.139315
$801$	0	0
$802$	10344.2	0.455446
$803$	−7707.91	−0.338738
$804$	0	0
$805$	1325.11	0.0580174
$806$	24793.7	1.08352
$807$	0	0
$808$	−23024.6	−1.00248
$809$	−23041.8	−1.00137	−0.500684	−	0.865630i	$-0.666918\pi$
$809$	−23041.8	−1.00137	−0.500684	+	0.865630i	$0.666918\pi$
$810$	0	0
$811$	34179.5	1.47991	0.739953	−	0.672659i	$-0.234848\pi$
$811$	34179.5	1.47991	0.739953	+	0.672659i	$0.234848\pi$
$812$	3088.05	0.133460
$813$	0	0
$814$	36844.8	1.58650
$815$	26546.0	1.14094
$816$	0	0
$817$	2202.04	0.0942959
$818$	−4476.86	−0.191357
$819$	0	0
$820$	−4250.36	−0.181011
$821$	−5743.98	−0.244173	−0.122087	−	0.992519i	$-0.538959\pi$
$821$	−5743.98	−0.244173	−0.122087	+	0.992519i	$0.538959\pi$
$822$	0	0
$823$	28540.8	1.20883	0.604416	−	0.796669i	$-0.293406\pi$
$823$	28540.8	1.20883	0.604416	+	0.796669i	$0.293406\pi$
$824$	25986.4	1.09864
$825$	0	0
$826$	−2607.48	−0.109838
$827$	−33133.5	−1.39319	−0.696594	−	0.717466i	$-0.745302\pi$
$827$	−33133.5	−1.39319	−0.696594	+	0.717466i	$0.745302\pi$
$828$	0	0
$829$	24606.9	1.03092	0.515460	−	0.856914i	$-0.327621\pi$
$829$	24606.9	1.03092	0.515460	+	0.856914i	$0.327621\pi$
$830$	13819.9	0.577947
$831$	0	0
$832$	−42326.9	−1.76373
$833$	−3026.82	−0.125898
$834$	0	0
$835$	21061.7	0.872897
$836$	1734.18	0.0717440
$837$	0	0
$838$	735.190	0.0303063
$839$	9141.86	0.376177	0.188088	−	0.982152i	$-0.439771\pi$
$839$	9141.86	0.376177	0.188088	+	0.982152i	$0.439771\pi$
$840$	0	0
$841$	1321.26	0.0541743
$842$	−30490.0	−1.24793
$843$	0	0
$844$	−5604.49	−0.228572
$845$	27572.5	1.12251
$846$	0	0
$847$	44024.6	1.78595
$848$	−18946.1	−0.767233
$849$	0	0
$850$	−8376.16	−0.338000
$851$	−2209.19	−0.0889897
$852$	0	0
$853$	−1471.84	−0.0590794	−0.0295397	−	0.999564i	$-0.509404\pi$
$853$	−1471.84	−0.0590794	−0.0295397	+	0.999564i	$0.509404\pi$
$854$	−30818.0	−1.23486
$855$	0	0
$856$	938.542	0.0374751
$857$	−35038.0	−1.39659	−0.698294	−	0.715811i	$-0.746057\pi$
$857$	−35038.0	−1.39659	−0.698294	+	0.715811i	$0.746057\pi$
$858$	0	0
$859$	6739.91	0.267710	0.133855	−	0.991001i	$-0.457264\pi$
$859$	6739.91	0.267710	0.133855	+	0.991001i	$0.457264\pi$
$860$	−827.036	−0.0327927
$861$	0	0
$862$	4943.15	0.195318
$863$	−265.457	−0.0104708	−0.00523538	−	0.999986i	$-0.501666\pi$
$863$	−265.457	−0.0104708	−0.00523538	+	0.999986i	$0.501666\pi$
$864$	0	0
$865$	9623.34	0.378270
$866$	−6989.62	−0.274269
$867$	0	0
$868$	2423.94	0.0947857
$869$	27651.9	1.07943
$870$	0	0
$871$	21965.9	0.854520
$872$	5031.17	0.195386
$873$	0	0
$874$	624.903	0.0241850
$875$	−25077.3	−0.968877
$876$	0	0
$877$	−46175.2	−1.77791	−0.888955	−	0.457995i	$-0.848568\pi$
$877$	−46175.2	−1.77791	−0.888955	+	0.457995i	$0.848568\pi$
$878$	29828.5	1.14654
$879$	0	0
$880$	26787.1	1.02613
$881$	31385.6	1.20024	0.600118	−	0.799912i	$-0.295120\pi$
$881$	31385.6	1.20024	0.600118	+	0.799912i	$0.295120\pi$
$882$	0	0
$883$	16931.3	0.645281	0.322641	−	0.946522i	$-0.395429\pi$
$883$	16931.3	0.645281	0.322641	+	0.946522i	$0.395429\pi$
$884$	4462.23	0.169775
$885$	0	0
$886$	−38316.8	−1.45291
$887$	−34774.3	−1.31635	−0.658177	−	0.752863i	$-0.728672\pi$
$887$	−34774.3	−1.31635	−0.658177	+	0.752863i	$0.728672\pi$
$888$	0	0
$889$	−34427.1	−1.29882
$890$	−21128.1	−0.795749
$891$	0	0
$892$	54.1557	0.00203281
$893$	−10600.5	−0.397237
$894$	0	0
$895$	10991.2	0.410496
$896$	17952.4	0.669363
$897$	0	0
$898$	−28386.2	−1.05485
$899$	20181.1	0.748695
$900$	0	0
$901$	18384.8	0.679786
$902$	−76845.4	−2.83666
$903$	0	0
$904$	−21295.8	−0.783505
$905$	−17716.6	−0.650739
$906$	0	0
$907$	−35925.5	−1.31520	−0.657601	−	0.753367i	$-0.728429\pi$
$907$	−35925.5	−1.31520	−0.657601	+	0.753367i	$0.728429\pi$
$908$	3535.29	0.129210
$909$	0	0
$910$	−26484.0	−0.964764
$911$	−48786.7	−1.77429	−0.887143	−	0.461495i	$-0.847313\pi$
$911$	−48786.7	−1.77429	−0.887143	+	0.461495i	$0.847313\pi$
$912$	0	0
$913$	−41575.4	−1.50706
$914$	−20345.2	−0.736279
$915$	0	0
$916$	−2595.02	−0.0936045
$917$	37692.5	1.35738
$918$	0	0
$919$	−18322.3	−0.657669	−0.328834	−	0.944388i	$-0.606656\pi$
$919$	−18322.3	−0.657669	−0.328834	+	0.944388i	$0.606656\pi$
$920$	−1879.89	−0.0673677
$921$	0	0
$922$	−19019.9	−0.679378
$923$	16811.2	0.599508
$924$	0	0
$925$	13791.1	0.490214
$926$	23402.3	0.830504
$927$	0	0
$928$	−8214.90	−0.290590
$929$	−30350.8	−1.07188	−0.535940	−	0.844256i	$-0.680043\pi$
$929$	−30350.8	−1.07188	−0.535940	+	0.844256i	$0.680043\pi$
$930$	0	0
$931$	−1409.67	−0.0496243
$932$	−3918.72	−0.137727
$933$	0	0
$934$	−46933.7	−1.64424
$935$	−25993.5	−0.909174
$936$	0	0
$937$	52314.3	1.82394	0.911971	−	0.410256i	$-0.134560\pi$
$937$	52314.3	1.82394	0.911971	+	0.410256i	$0.134560\pi$
$938$	−12906.0	−0.449248
$939$	0	0
$940$	3981.31	0.138145
$941$	−38821.1	−1.34488	−0.672440	−	0.740152i	$-0.734753\pi$
$941$	−38821.1	−1.34488	−0.672440	+	0.740152i	$0.734753\pi$
$942$	0	0
$943$	4607.60	0.159114
$944$	3160.48	0.108967
$945$	0	0
$946$	−14952.6	−0.513901
$947$	−4992.59	−0.171317	−0.0856586	−	0.996325i	$-0.527299\pi$
$947$	−4992.59	−0.171317	−0.0856586	+	0.996325i	$0.527299\pi$
$948$	0	0
$949$	−9236.88	−0.315955
$950$	−3901.01	−0.133227
$951$	0	0
$952$	−20999.8	−0.714923
$953$	18808.2	0.639304	0.319652	−	0.947535i	$-0.396434\pi$
$953$	18808.2	0.639304	0.319652	+	0.947535i	$0.396434\pi$
$954$	0	0
$955$	11309.0	0.383196
$956$	6264.71	0.211940
$957$	0	0
$958$	−4538.99	−0.153077
$959$	35871.5	1.20788
$960$	0	0
$961$	−13950.0	−0.468262
$962$	44153.5	1.47980
$963$	0	0
$964$	−81.5084	−0.00272324
$965$	22289.8	0.743559
$966$	0	0
$967$	113.323	0.00376857	0.00188428	−	0.999998i	$-0.499400\pi$
$967$	113.323	0.00376857	0.00188428	+	0.999998i	$0.499400\pi$
$968$	−62456.3	−2.07378
$969$	0	0
$970$	−8302.23	−0.274813
$971$	−18935.8	−0.625827	−0.312913	−	0.949782i	$-0.601305\pi$
$971$	−18935.8	−0.625827	−0.312913	+	0.949782i	$0.601305\pi$
$972$	0	0
$973$	45748.3	1.50732
$974$	−6801.78	−0.223761
$975$	0	0
$976$	37353.9	1.22507
$977$	−9255.46	−0.303079	−0.151540	−	0.988451i	$-0.548423\pi$
$977$	−9255.46	−0.303079	−0.151540	+	0.988451i	$0.548423\pi$
$978$	0	0
$979$	63561.3	2.07500
$980$	529.441	0.0172575
$981$	0	0
$982$	−19703.0	−0.640272
$983$	−21559.8	−0.699544	−0.349772	−	0.936835i	$-0.613741\pi$
$983$	−21559.8	−0.699544	−0.349772	+	0.936835i	$0.613741\pi$
$984$	0	0
$985$	−32938.9	−1.06550
$986$	−21828.0	−0.705016
$987$	0	0
$988$	2078.18	0.0669188
$989$	896.548	0.0288257
$990$	0	0
$991$	35161.9	1.12710	0.563550	−	0.826082i	$-0.309435\pi$
$991$	35161.9	1.12710	0.563550	+	0.826082i	$0.309435\pi$
$992$	−6448.23	−0.206383
$993$	0	0
$994$	−9877.31	−0.315180
$995$	−40227.9	−1.28172
$996$	0	0
$997$	−48076.0	−1.52716	−0.763582	−	0.645711i	$-0.776561\pi$
$997$	−48076.0	−1.52716	−0.763582	+	0.645711i	$0.776561\pi$
$998$	24599.0	0.780227
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.c.1.5		7
3.2	odd	2		177.4.a.b.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.b.1.3	✓	7	3.2	odd	2
531.4.a.c.1.5		7	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61892 q^{2} -1.14126 q^{4} +7.96684 q^{5} +16.8751 q^{7} -23.9402 q^{8} +O(q^{10})\)	\(q+2.61892 q^{2} -1.14126 q^{4} +7.96684 q^{5} +16.8751 q^{7} -23.9402 q^{8} +20.8645 q^{10} -62.7682 q^{11} -75.2191 q^{13} +44.1946 q^{14} -53.5675 q^{16} +51.9804 q^{17} +24.2087 q^{19} -9.09222 q^{20} -164.385 q^{22} +9.85642 q^{23} -61.5295 q^{25} -196.993 q^{26} -19.2589 q^{28} -160.344 q^{29} -125.861 q^{31} +51.2329 q^{32} +136.132 q^{34} +134.441 q^{35} -224.138 q^{37} +63.4006 q^{38} -190.728 q^{40} +467.472 q^{41} +90.9608 q^{43} +71.6348 q^{44} +25.8132 q^{46} -437.881 q^{47} -58.2301 q^{49} -161.141 q^{50} +85.8445 q^{52} +353.688 q^{53} -500.064 q^{55} -403.994 q^{56} -419.929 q^{58} -59.0000 q^{59} -697.325 q^{61} -329.620 q^{62} +562.715 q^{64} -599.258 q^{65} -292.026 q^{67} -59.3231 q^{68} +352.091 q^{70} -223.496 q^{71} +122.800 q^{73} -586.998 q^{74} -27.6284 q^{76} -1059.22 q^{77} -440.541 q^{79} -426.763 q^{80} +1224.27 q^{82} +662.365 q^{83} +414.119 q^{85} +238.219 q^{86} +1502.68 q^{88} -1012.63 q^{89} -1269.33 q^{91} -11.2487 q^{92} -1146.77 q^{94} +192.867 q^{95} -397.912 q^{97} -152.500 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8}+O(q^{10}) \) 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8	\( 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8} - 71 q^{10} + 5 q^{11} - 67 q^{13} + 65 q^{14} - 94 q^{16} + 23 q^{17} - 176 q^{19} + 207 q^{20} - 704 q^{22} + 218 q^{23} - 183 q^{25} - 58 q^{26} - 938 q^{28} - 168 q^{29} - 604 q^{31} + 448 q^{32} - 610 q^{34} + 336 q^{35} - 505 q^{37} + 453 q^{38} - 1080 q^{40} + 265 q^{41} - 493 q^{43} - 504 q^{44} + 381 q^{46} + 244 q^{47} + 770 q^{49} - 1639 q^{50} + 160 q^{52} - 686 q^{53} - 116 q^{55} - 2190 q^{56} + 1584 q^{58} - 413 q^{59} - 838 q^{61} - 286 q^{62} + 205 q^{64} - 490 q^{65} - 1504 q^{67} - 3047 q^{68} + 1530 q^{70} + 1267 q^{71} - 666 q^{73} - 528 q^{74} - 64 q^{76} - 1109 q^{77} - 2741 q^{79} - 1213 q^{80} + 953 q^{82} + 2025 q^{83} - 1274 q^{85} - 4394 q^{86} - 1639 q^{88} - 616 q^{89} - 2415 q^{91} - 218 q^{92} + 900 q^{94} - 2554 q^{95} - 1298 q^{97} + 172 q^{98}+O(q^{100}) \) 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8 - 71 * q^10 + 5 * q^11 - 67 * q^13 + 65 * q^14 - 94 * q^16 + 23 * q^17 - 176 * q^19 + 207 * q^20 - 704 * q^22 + 218 * q^23 - 183 * q^25 - 58 * q^26 - 938 * q^28 - 168 * q^29 - 604 * q^31 + 448 * q^32 - 610 * q^34 + 336 * q^35 - 505 * q^37 + 453 * q^38 - 1080 * q^40 + 265 * q^41 - 493 * q^43 - 504 * q^44 + 381 * q^46 + 244 * q^47 + 770 * q^49 - 1639 * q^50 + 160 * q^52 - 686 * q^53 - 116 * q^55 - 2190 * q^56 + 1584 * q^58 - 413 * q^59 - 838 * q^61 - 286 * q^62 + 205 * q^64 - 490 * q^65 - 1504 * q^67 - 3047 * q^68 + 1530 * q^70 + 1267 * q^71 - 666 * q^73 - 528 * q^74 - 64 * q^76 - 1109 * q^77 - 2741 * q^79 - 1213 * q^80 + 953 * q^82 + 2025 * q^83 - 1274 * q^85 - 4394 * q^86 - 1639 * q^88 - 616 * q^89 - 2415 * q^91 - 218 * q^92 + 900 * q^94 - 2554 * q^95 - 1298 * q^97 + 172 * q^98

Introduction

L-functions

Modular forms

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Database

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